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The hosts nearly shot themselves in the foot within the first few minutes as Ashley Williams played a poor back-pass towards Lukasz Fabianski that was scrambled out for a Liverpool corner, but Martin Skrtel went on to head over from the set piece.
Brendan Rodgers's charges continued to press and came close in the seventh minute when Adam Lallana combined with Raheem Sterling to set up Daniel Sturridge, the former Chelsea and Manchester City man belting his effort just wide.
Swansea rallied round and threatened on the quarter-hour mark, Joe Allen having to time his tackle on Bafetimbi Gomis perfectly before Ki Sung-yueng headed back towards goal, where Skrtel headed clear.
Former Liverpool man Jonjo Shelvey also came close to opening the scoring, firing just wide from a short corner, but the visitors were able to see out the remainder of the half unscathed.
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The second half started with plenty of physical intent but little in the way of attacking, with the first concrete effort coming just before the hour when Philippe Coutinho's low effort from close range forced a good diving save from Fabianski.
Liverpool improved as the game went on and succeeded in breaking the deadlock 66 minutes in, Jordan Henderson racing onto to a through-ball flicked into his path by Sturridge.
As a Swans defender attempted to clear the ball out of danger, the ball rebounded off Henderson's shin, over Fabianski and into the net.
From then on it was all Liverpool as the hosts were unable to work their way back into the game and were lucky not to concede more as Coutinho and Sterling came close in the latter stages.
The Reds remain fifth in the Premier League table, while Swansea sit ninth.