James Milner proved the hero for the Reds, netting home a late penalty and pilling more pressure on Swansea's embattled manager Francesco Guidolin with his side having now gone six Premier League matches without a win.
Despite Liverpool coming into the match as strong favourites it was the home team who started the game on the front foot.
The first goal came after the Swans defended the Reds first concentrated attack well, breaking quickly before winning a corner after a beautiful attacking run by Gylfi Sigurðsson.
Both sides struggled to contain the corner before a scramble led to Mike van der Hoorn knocking the ball onto Leroy Fer, who touched the ball into the back of the net past a helpless Loris Karius to put Swansea up 1-0 in the eighth minute.
The setbacks for Liverpool continued with an in-form Adam Lallana forced off with an injury in the 23rd minute, giving Daniel Sturridge the chance to stake a claim for a position in the starting eleven.
Sigurdsson's set-piece delivery exposed the Reds once more, but Borja's off-target header came as the linesman's flag went up for offside.
Coutinho had a chance to even the scores in the 27th minute after finding himself with space and a chasing Sturridge but misplaced his pass straight to Karius.
The second half started off as a much more evenly contested affair.
After a number of chances at both ends Roberto Firmino equaled the score in the 54th minute, captialising off of blocked free kick that rebounded to Jordan Henderson who chipped the ball beautifully to the Brazilian, nodding in a header at the far post.
From that point Liverpool took control of the match despite Swansea's best efforts.
After a prolonged period of holding out the Reds' attack, Angel clumsily challenged Firmino in the box, pushing the attacker down with penalty being awarded.
James Milner calmly powered the ball into the middle of the net past Swans keeper Lukasz Fabianski in the 84th minute, converting his fourth penalty of the season and putting his side on course to take the points at Liberty Stadium in a 2-1 win.