Top scorer Michu was left on the bench, along with regular starters Ashley Williams, Angel Rangel, Routledge and Nathan Dyer.
That saw winger Roland Lamah make his first league start for the Welsh side, while Israel striker Itay Shechter was asked to lead the line.
Suarez tested Michel Vorm, firing straight at the Swans goalkeeper, while Stewart Downing and Glen Johnson also missed chances as the Reds retained a firm grip on the game.
Did you know?
Liverpool had failed to score in any of the three previous Premier League meetings with Swansea.
This is the first time in the Premier League era in which two different Liverpool players have converted a penalty in the same league match
Suarez was twice brought down on the edge of the area, but first Gerrard and then Suarez were unable to find the target with free-kicks.
Sturridge wasted a golden chance, getting under his header six yards out after Downing had beaten Ben Davies down the Liverpool right.
Coutinho then missed the chance to open his account in front of the Kop, picking up a loose ball after Sturridge had driven into the Swans area but putting the ball just wide.
The breakthrough came after Agustien appeared to clip Suarez's ankle in the area, referee Howard Webb pointing to the penalty spot after assistant Mike Mullarkey flagged for the offence.
Gerrard, who had missed a penalty in the home loss to West Bromwich Albion, held his nerve and just beat the plunging Vorm low to the keeper's right.
That was the first goal Swansea had conceded to Liverpool in their three previous Premier League encounters, but a second, third and fourth followed in the opening 11 minutes of the second half.
First Suarez released Coutinho and the Brazilian surged past two defenders before beating Vorm with a cool finish.
Coutinho was then just wide with another attempt before Sturridge, Suarez and Enrique combined with a mesmerising move down the left.
Laudrup sorry for Swans 'nightmare'
The ball was pinged between bemused Swansea defenders to set up Enrique, drifting inside, to power his shot into the roof of the net from close range.
Swansea's early resolve and discipline had dissolved under the flurry of goals and Suarez was able to jink past Kyle Bartley and Garry Monk before picking his spot for Liverpool's fourth.
Routledge had come off the bench for Swansea but unfortunately for the visitors the winger conceded a penalty for handball when he challenged Suarez for a lofted pass.
Gerrard generously gave the spot-kick to Sturridge, but the result was the same as the former Chelsea man slammed the ball past Vorm.
Liverpool were playing their third game in seven days, having lost 2-0 on Monday at home against West Brom before losing in Russia on Thursday in the first leg of their Europa League last 32 game with Zenit St Petersburg.
But this match ended on a sour note for the hosts as striker Fabio Borini, who only came on as a 78th minute substitute, landed badly in a challenge and dislocated a shoulder, an injury that will rule him out for the rest of the season.
The Italian, signed for £10.5m from Roma in July, left the field screaming in agony as the Reds ended the match with 10 men after already making their three substitutions.
Liverpool's victory saw them climb above Swansea into seventh place but the Swans must regroup for their Wembley encounter with Bradford next Sunday as they bid to claim their first major trophy and a place in next season's Europa League.
88:01 Wayne Routledge challenges Glen Johnson unfairly and gives away a free kick. Direct free kick taken by Steven Gerrard.
83:10 Effort from 18 yards by Wayne Routledge. Save made by Jose Reina.
81:27 Substitution Angel Rangel is brought on as a substitute for Itay Shechter.
81:27 Glen Johnson has an effort at goal from just inside the box that misses to the left of the target.
80:31 Short corner worked by Stewart Downing.
80:22 The ball is swung over by Sanchez Jose Enrique, Dwight Tiendalli gets a block in.
78:03 Steven Gerrard takes a outswinging corner.
77:48 Outswinging corner taken by Steven Gerrard, clearance made by Garry Monk.
77:04 Substitution Fabio Borini comes on in place of Luis Suarez.
75:20 Joe Allen takes the direct free kick.
75:20 Substitution Pablo Hernandez goes off and Nathan Dyer comes on.
75:20 Booking Pablo Hernandez booked for unsporting behaviour.
75:17 Unfair challenge on Daniel Agger by Pablo Hernandez results in a free kick.
74:12 A cross is delivered by Jordan Henderson.
73:20 Wayne Routledge fouled by Glen Johnson, the ref awards a free kick. Jonathan de Guzman crosses the ball from the free kick
right-footed from left channel, comfortable save by Jose Reina.
72:41 Shot on goal by Daniel Sturridge from just inside the area clears the crossbar.
71:54 Substitution Leiva Lucas leaves the field to be replaced by Joe Allen.
70:52 Sanchez Jose Enrique provided the assist for the goal.
70:52 GOAL - Daniel Sturridge:Liverpool 5 - 0 Swansea Penalty taken left-footed by Daniel Sturridge and scored. Liverpool 5-0 Swansea.
69:59 Wayne Routledge is penalised for handball and concedes a .
67:37 The ball is sent over by Luis Suarez.
67:00 Corner taken right-footed by Luis Suarez, Effort from 20 yards by Steven Gerrard. Blocked by Kyle Bartley.
66:43 Effort on goal by Daniel Sturridge from 18 yards. Save made by Michel Vorm.
64:50 Glen Johnson has an effort at goal from long distance that misses to the right of the goal.
63:31 Substitution Roland Lamah leaves the field to be replaced by Wayne Routledge.
62:55 Daniel Sturridge takes a shot. Save by Michel Vorm. Shot by Luis Suarez from 20 yards. Blocked by Kyle Bartley.
62:14 Free kick awarded for an unfair challenge on Glen Johnson by Ben Davies. Glen Johnson takes the direct free kick.
59:25 Substitution Jordan Henderson is brought on as a substitute for Phillippe Coutinho.
55:55 Stewart Downing provided the assist for the goal.
55:55 GOAL - Luis Suarez:Liverpool 4 - 0 Swansea Luis Suarez gets on the score sheet with a goal from just inside the area to the bottom left corner of the goal. Liverpool
54:48 A cross is delivered by Dwight Tiendalli, Daniel Agger manages to make a clearance.
53:44 Corner taken by Steven Gerrard, Headed effort from inside the area by Daniel Agger misses to the right of the target.
53:35 The ball is swung over by Luis Suarez, Ben Davies gets a block in.
52:44 The assistant referee flags for offside against Daniel Sturridge. Dwight Tiendalli restarts play with the free kick.
51:06 Free kick awarded for an unfair challenge on Itay Shechter by Daniel Agger. The free kick is delivered left-footed by Roland
Lamah from right wing, save by Jose Reina.
49:44 The assist for the goal came from Daniel Sturridge.
49:44 GOAL - Sanchez Jose Enrique:Liverpool 3 - 0 Swansea Sanchez Jose Enrique gets on the score sheet with a goal from inside the box high into the middle of the goal. Liverpool 3-0
47:56 Foul by Jonathan de Guzman on Leiva Lucas, free kick awarded. Free kick taken by Sanchez Jose Enrique.
46:54 Luis Suarez produces a cross, Phillippe Coutinho produces a right-footed shot from deep inside the penalty box which goes
wide of the left-hand post.
45:15 Luis Suarez provided the assist for the goal.
45:15 GOAL - Phillippe Coutinho:Liverpool 2 - 0 Swansea Phillippe Coutinho finds the net with a goal from just inside the area low into the middle of the goal. Liverpool 2-0 Swansea.
45:01 The referee gets the second half underway.
45:00+1:15 The ref blows to end the first period.
45:00+0:40 Garry Monk challenges Luis Suarez unfairly and gives away a free kick. Free kick taken by Jose Reina.
Correction - 43:20 Sanchez Jose Enrique challenges Itay Shechter unfairly and gives away a free kick. Jonathan de Guzman sends in a cross, Garry
Monk takes a shot. Glen Johnson makes a clearance.
43:20 Sanchez Jose Enrique challenges Itay Shechter unfairly and gives away a free kick. Jonathan de Guzman crosses the ball, Garry
Monk takes a shot. Glen Johnson makes a clearance.
41:14 Jonathan de Guzman takes a shot. Save by Jose Reina.
39:27 The ball is sent over by Daniel Sturridge, save by Michel Vorm.
38:59 Daniel Sturridge takes a shot from 12 yards. Kyle Bartley makes a clearance.
38:33 Phillippe Coutinho has a drilled shot. Kyle Bartley gets a block in.
38:04 Stewart Downing takes a shot. Ben Davies gets a block in.
36:34 The ball is delivered by Glen Johnson.
35:56 Shot by Steven Gerrard. Garry Monk gets a block in.
35:31 Shot by Leiva Lucas from 18 yards. Kyle Bartley gets a block in.
33:55 The assist for the goal came from Luis Suarez.
33:55 GOAL - Steven Gerrard:Liverpool 1 - 0 Swansea Steven Gerrard scores a placed penalty. Liverpool 1-0 Swansea.
32:36 Kemy Agustien concedes a Penalty for a foul on Luis Suarez.
31:52 Effort from 18 yards by Steven Gerrard. Blocked by Leon Britton. Corner taken by Steven Gerrard from the left by-line, Ben
Davies makes a clearance.
30:06 The ball is crossed by Pablo Hernandez, save by Jose Reina.
28:57 Leiva Lucas has an effort at goal from 30 yards. Save by Michel Vorm.
27:27 Itay Shechter is caught offside. Free kick taken by Sanchez Jose Enrique.
25:28 Daniel Sturridge takes a shot. Save made by Michel Vorm.
24:41 Leiva Lucas concedes a free kick for a foul on Jonathan de Guzman. Leon Britton takes the free kick.
24:01 Shot by Phillippe Coutinho from deep inside the penalty area misses to the left of the goal.
22:34 The assistant referee signals for offside against Itay Shechter. Indirect free kick taken by Leiva Lucas.
22:02 The ball is swung over by Stewart Downing, Daniel Sturridge has a header from deep inside the six-yard box and clears the
21:32 Centre by Pablo Hernandez, Jamie Carragher makes a clearance.
19:20 Luis Suarez produces a right-footed shot from just outside the penalty area that goes harmlessly over the target.
16:53 Dwight Tiendalli challenges Luis Suarez unfairly and gives away a free kick. Luis Suarez has an effort direct from the free
14:41 Free kick awarded for an unfair challenge on Luis Suarez by Leon Britton. Steven Gerrard produces a strike on goal direct
from the free kick. Luis Suarez takes a shot from just inside the penalty box which clears the bar.
13:44 Shot from 18 yards by Stewart Downing.
13:17 Steven Gerrard takes a inswinging corner to the near post, clearance made by Itay Shechter.
13:03 Daniel Sturridge takes a shot. Kyle Bartley gets a block in.
12:04 Inswinging corner taken by Stewart Downing from the right by-line, Headed effort from inside the area by Glen Johnson misses
to the left of the target.
11:55 The ball is crossed by Glen Johnson, clearance made by Garry Monk.
10:39 Stewart Downing produces a left-footed shot from just outside the box that misses to the right of the goal.
9:41 Glen Johnson fouled by Itay Shechter, the ref awards a free kick. Leiva Lucas takes the free kick.
7:53 Corner taken by Jonathan de Guzman from the right by-line, Roland Lamah takes a shot. Glen Johnson gets a block in.
7:02 Leiva Lucas gives away a free kick for an unfair challenge on Kemy Agustien. Kyle Bartley takes the free kick.
5:26 Shot by Luis Suarez from 18 yards. Michel Vorm makes a save.
3:03 Unfair challenge on Dwight Tiendalli by Phillippe Coutinho results in a free kick. Free kick taken by Michel Vorm.
1:41 Inswinging corner taken from the left by-line by Steven Gerrard, Effort from inside the area by Stewart Downing misses to
the right of the target.
1:26 Shot by Phillippe Coutinho. Kyle Bartley gets a block in.
1:02 Corner taken short by Steven Gerrard.
0:00 The game gets underway.
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