After Fernando Llorente scored in the 48th and 52nd minutes to give the Swans a 2-0 lead, Liverpool responded with Robert Firmino's double (55', 69').
The Reds halved the deficit through Roberto Firmino's header before the Brazil global netted a sumptuous volley to level the Premier League clash with 21 minutes remaining.
Swansea pulled off a shocking victory over Liverpool in a brilliant game at Anfield.
With the result, Liverpool could be 10 points off the top if Chelsea beats Hull on Sunday.
Yet for all of Liverpool's domination of possession, these were rare openings and the nearest to a first-half goal was when Carroll, signed from Tottenham Hotspur in midweek, saw his shot from a narrow angle deflect off Dejan Lovren and strike the post.
The win lifted Swansea out of the drop zone and into 17th place pending the outcome of later matches, while Liverpool remain third and could lose more ground before the weekend is over.
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Roberto Firmino struck twice to draw Liverpool level, but the hosts switched off again as Gylfi Sigurdsson secured lowly Swansea's first ever league win at Anfield.
"It's a big shock today because before the game I'm sure they'd have been seeing it as a three-point game".
"One defeat doesn't make you a bad team, we've still got a lot of confidence in the squad. Then we had the punch that we missed a little bit in the first half, not only when we scored the goals, also when we created the chances".
'[For the] second goal, we can defend the cross, it's why you should avoid crosses. The reaction on the long ball was not good.
That would leave Liverpool's title hopes hanging by a slender thread and Klopp challenged his players to prove they can bounce back quickly after such an embarrassing loss. But as soon as there is any wear and tear they can struggle to cope with the demands needed to win the league, rather than just light it up when all is going well. I think it started with a long ball - that was not a real surprise, they played a few of these. That is a very important part of the game and was very disappointing today. So, we're very disappointed but the games come thick and fast and we need to be ready for the next one. But it was Swansea that hit Liverpool in the second half once more, as Gylfi Sigurdsson scored the victor.