Jamie Carragher's farewell Liverpool appearance before retirement ended with a comfortable win over relegated QPR.
The Reds were denied an early goal when Philippe Coutinho's header following a corner seemed to cross the line.
QPR then had a penalty appeal turned down for a Lucas Leiva trip on striker Loic Remy before Brazilian Coutinho netted with a 25-yard strike.
Carragher a true legend - Rodgers
Carragher almost scored a dream goal
on his departure
only for his long-range shot to bounce back off the post.
The central defender was substituted to a huge ovation late on, with Anfield bursting into a rendition of 'we all dream of a team of Carraghers' on his exit.
Earlier, there had been a huge roar as the former England international walked out for his 737th club appearance. He was given a guard of honour by both sets of players, with the Kop paying its own tribute by displaying a mosaic of his initials and squad number.
The 35-year-old went on to bow out of the game with a clean sheet, while relegated QPR said their own goodbye to the Premier League with a defeat which left them
bottom of the table.
As the clock ticked down on Carragher's time at Anfield, Jordon Ibe was handed his first-team debut by Brendan Rodgers and the 17-year-old winger impressed by setting up his side's goal in the 23rd minute.
After Coutinho's goal that never was and the rejection of Remy's penalty appeal, Ibe jinked in from the left and laid the ball off for playmaker Coutinho to send an unerring strike into the bottom corner.
Carragher escaped a booking from referee Martin Atkinson when he tried to punch in a Coutinho cross at the far post as the Reds threatened a second.
Alistair MannBBC Match of the Day commentator
"Each one of Jamie Carragher's 737 appearances were filled with determination, commitment and reliability. Today was the day to say farewell. He didn't score - his long-range shot hitting the post - but the clean sheet will mean as much as a goal to him."
Bobby Zamora also had a chance for the visitors only for the striker to fail to connect with an attempted side-footed effort after Martin Skrtel could not prevent a low cross from reaching him.
Liverpool, though, were the team with a firm grip on the match and QPR keeper Rob Green smartly kept out deflected strikes from Jose Enrique and Glen Johnson, while an angled Ibe shot went wide of his far post.
The Kop were willing Carragher to score and he almost did so in spectacular style only for his fierce 30-yard shot to hit the post.
Liverpool's victory would have been more emphatic had it not been for Green but the final word inevitably belonged to Carragher, who was loudly cheered when he was substituted shortly before the end.
Liverpool manager Brendan Rodgers:
"The win marks a terrific end to the season for us. We would love the season to continue as we have found our feet and form. [But] we've got work to do during the summer in terms of reinforcing the group.
"Carra is a true great. This is a guy, who after 737 games, is a true legend of this club. His performance was immaculate. I only wish for him and his family that his strike would have gone in. He gets credit for defensive qualities but he is also a great technical player.
"It has been a pleasure working with him. My only regret is that I would have liked to have had many more years and games working with him."
QPR boss Harry Redknapp:
"We've been close in a lot of games but lacked quality to turn those close games around.
"We didn't have the quality to stay in the Premier League. We need to improve that in the summer with lads who want to work hard and do well for QPR.
"We've got no divine right to come straight back up. We'll have to work hard. I've inherited a group with some good professionals and also with people who are not so good.
"That has been the problem. Lots of them can't get up early enough to get to training on time."
45:00+0:30 Phillippe Coutinho takes a shot. Rob Green makes a save.
44:18 Shot from just outside the area by Daniel Sturridge goes over the bar.
42:52 Free kick awarded for a foul by Jordon Ibe on Stephane Mbia. Centre by Andros Townsend, Headed effort from deep inside the
penalty area by Stephane Mbia goes harmlessly over the crossbar.
42:14 Effort from inside the area by Sanchez Jose Enrique misses to the right of the goal.
39:05 Free kick awarded for a foul by Jamie Carragher on Bobby Zamora. Andros Townsend crosses the ball from the free kick left-footed
from right channel, save made by Rob Green.
38:28 Andros Townsend has an effort at goal from outside the penalty box missing to the wide right of the target.
Correction - 35:56 Jordan Henderson takes a shot. Blocked by Nedum Onuoha.
35:56 Jordan Henderson takes a shot. Nedum Onuoha gets a block in.
34:29 Loic Remy produces a left-footed shot from just outside the box that misses to the right of the net. Effort on goal by Glen
Johnson from just outside the area goes harmlessly over the target.
33:17 Andros Townsend challenges Glen Johnson unfairly and gives away a free kick. Stewart Downing takes the free kick, Jamie Carragher
takes a shot. Save by Rob Green.
31:54 Foul by Clint Hill on Phillippe Coutinho, free kick awarded. Shot on goal comes in from Jordan Henderson from the free kick.
31:02 Booking The referee shows Clint Hill a yellow card.
27:22 Foul by Phillippe Coutinho on Michael Harriman, free kick awarded. Nedum Onuoha restarts play with the free kick.
26:39 Free kick awarded for a foul by Jamie Carragher on Rob Green. Rob Green takes the direct free kick.
25:57 Corner taken by Phillippe Coutinho.
22:15 Jordon Ibe provided the assist for the goal.
22:15 GOAL - Phillippe Coutinho:Liverpool 1 - 0 QPR Phillippe Coutinho finds the net with a goal from outside the area to the bottom left corner of the goal. Liverpool 1-0 QPR.
20:56 Stewart Downing takes a shot. Save by Rob Green.
17:24 Phillippe Coutinho takes a shot. Shaun Derry gets a block in. Corner taken by Stewart Downing from the right by-line to the
near post, Glen Johnson takes a shot. Michael Harriman manages to make a clearance.
16:53 Stephane Mbia has an effort at goal from outside the penalty area which goes wide right of the goal.
15:42 Stewart Downing takes a short corner.
14:21 The official flags Phillippe Coutinho offside. Rob Green restarts play with the free kick.
13:01 Corner taken right-footed by Phillippe Coutinho, save by Rob Green. Inswinging corner taken from the right by-line by Stewart
Downing, Ji-Sung Park makes a clearance.
11:13 Andros Townsend takes a shot. Jose Reina makes a save.
7:44 Nedum Onuoha restarts play with the free kick.
7:44 Booking Jordan Henderson is shown a yellow card.
7:36 Jordan Henderson gives away a free kick for an unfair challenge on Ji-Sung Park.
5:22 Ji-Sung Park takes a shot. Blocked by Martin Skrtel.
1:54 Corner taken right-footed by Phillippe Coutinho, save by Rob Green.
Correction - 0:55 Phillippe Coutinho takes a shot. Inswinging corner taken by Stewart Downing from the right by-line, Phillippe Coutinho takes
a shot. Corner taken right-footed by Phillippe Coutinho, Rob Green makes a save. Blocked by Clint Hill. Blocked by Nedum Onuoha.
0:55 Phillippe Coutinho takes a shot. Inswinging corner taken from the right by-line by Stewart Downing, Phillippe Coutinho takes
a shot. Corner taken by Phillippe Coutinho, save by Rob Green.
0:00 The referee gets the match started.
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