So, a Liverpool win increases the pressure on Garry Monk at Swansea, whose side are just four points above the relegation zone.
Swansea manager Garry Monk told BBC Sport: "I am gutted for my players as they were excellent. It was a very soft penalty and it is a shame for the the players as they were excellent today. I have no complaints over the performance.
"I did think penalties had to be deliberate. Neil's back is turned. It was very soft.
"We didn't look like a team in trouble. We are not going to make excuses, I know we need results but if we play like that it is a given.
"It was an excellent performance which deserved much more than it got.
"I take full responsibility for this period. You can see players are working hard to put it right."
Swansea boss Garry Monk is "gutted" to have lost and calls the decision to award a penalty for the goal "incredibly soft".
Full quotes to come, in possibly his last post-match interview as Swansea boss.
Liverpool manager Jurgen Klopp told BBC Sport: "It was really difficult for both teams, the wind was in the stadium more than the stands.
"Swansea defended very well. They had some problems in the last few games but they were more compact today and showed respect for our last few games.
"We made this one goal. It is different [to 4-1 win over Man City] but to get results in a row you have to fight for them. I am proud of the lads. It is a good result in difficult circumstances.
"They did not have that big chances. We defended really well. What Jordan Ibe did in tactical sense with his defending was a big step for him and us."
Jurgen Klopp says he is "really proud" of the result and laughs as he is told goalkeeper Simon Mignolet didn't have a save to make.
Full quotes to comes.
Liverpool captain James Milner: "It wasn't the prettiest of games. It was important to win.
"The win last week against Manchester City would have been a waste if we had not won.
"It was tough conditions for the match. I don't think it was ever going to be the best game to watch and it was all about getting the points.
"It was brilliant to see them (Henderson and Sturridge) out there as they are big misses when they are out.
"There is a long way to go but shows what a few wins can do."
Fourth-placed Arsenal drop points which means Liverpool are just four points behind the Gunners now. Can the Red challenge for a Champions League place? They're right in the running for it now.
Daniel Sturridge and Jordan Henderson back from injuries and some solid additions in the transfer window will make them a force to be reckoned with.
A stunning, attacking display last weekend and grinding out a win this week. Positive stuff for Jurgen Klopps' side.
Here we go...
Well done Liverpool. They take the points and jump up to sixth. Swansea now 15th and worrying times for the Welsh side.
Liverpool clear the corner and are second away from the three points...
CORNER! Last chance for Swansea. Lukasz Fabianski asks if he can go up but his manager says no!
One last chance for Swansea to get the ball into the Liverpool box...
Kolo, Kolo Kolo, Kolo Kolo, Kolo Toure Toure comes on for the last few seconds in place of Jordon Ibe who gets a pat on the back from his boss.
Liverpool goalkeeper Simon Mignolet booked for time wasting. He won't mind that one.
Awful corner, it really is a horrible one when in need of a goal. Gylfi Sigurdsson's corner is too long and sails out of play. Frustrating for the players, for the fans and for the boss.
Is there a twist in the tale? Not long to go.
Swansea work an opening and Bafetimbi Gomis's long-range effort deflects wide. CORNER!!
Liverpool have done really well to close out this game. Excellent game management which was lacking under Brendan Rodgers at times. The 3-3 draw at Crystal Palace a few seasons ago springs to mind.
FIVE minutes of added time.
The excellent Emre Can attempts a through ball but it runs straight to Swansea goalkeeper Lukasz Fabianski. It's a long way away from the away side's intended target at the moment. Garry Monk urging his troops on. Final few minutes of his tenure? This will make it one win in the last 10.
Liverpool break clear through but Adam Lallana gives the ball away. They manage to get it back and are happy to keep control of it with the clock tick tocking towards a conclusion.
Swansea well on top now but goalkeeper Simon Mignolet still hasn't been made to work. Can they fashion a chance late on?
Time running out for them and maybe boss Garry Monk's tenure too.
Nope, wasted by the visitors. A bit of pinball action before centre-back Ashley Williams heads wide. He can to trudge all the way back down the pitch. Into the last 10.
A coming together in the box between James Miner and Kyle Bartley but no penalty!
Again, it's not the best of deliveries and Liverpool can get the ball away. It's coming back though and ANOTHER Swansea corner. Can they make anything from it?
Swansea pressure! They force a corner and centre-back Ashley Williams make his way up the field..
A lack of communication sees two Liverpool defenders bump into each other. Jefferson Montero can run at the Reds defence for the first time and is almost brought down in the box by Jordan Henderson.
Would have been interesting if he went down there, the ref could easily have given his second spot-kick of the match.
Skipper Jordan Henderson right in the thick of it. Picks out goalscorer James Milner on the left, whose cross/shot dribbles wide. Daniel Sturridge almost gets on the end of it, but he doesn't want to exert himself too much after returning from injury.
What a character!
I called for Jefferson Montero earlier and have been granted my wish. The Ecuadorian on for Wayne Routledge and will look to run at a tiring Liverpool back line.
Here he is, Liverpool striker Daniel Sturridge participates for the first time since 4 October in the Merseyside derby against Everton. Christian Benteke is the man hooked.
He is immediately into the action with a few stepovers on the right hand side of the area, but doesn't get injured.
Goodness me! Kyle Bartley with a scything challenge on James Milner and goes into the book. That was a rough tackle!!
Like for like change for Swansea as the battered and bruised Eder comes off for Bafetimbi Gomis. Game changer?
Right then Swansea, what have you got? What can you show?
Swansea striker Eder has been in the wars today and is down again, this time after a chop from Liverpool defender Martin Skrtel. A deserved yellow for the Slovakian.