Adam Lallana and Tom Carroll had the best chances of a lacklustre opening half, but that all changed after the restart as the game burst into life.
Meanwhile, Swansea come to the game on the back of a 4-0 thrashing at the hands of Arsenal despite starting the game on the front foot.
Spain striker Fernando Llorente scored twice in quick succession after the break for the Swans and although Robert Firmino's double hauled Liverpool level, Gylfi Sigurdsson slotted in a 74th-minute victor to inflict Liverpool's first home league defeat of the season and lift Swansea out of the bottom three.
Liverpool manager Jurgen Klopp's frustration grew as Swansea went ahead with little more than two minutes of the second half played.
And when given an opportunity to single out individual teammates for the overall lacking performance and throw them under the bus, the skipper took a hard pass.
Both Can and Jordan Henderson had been earmarked as Liverpool's midfield pairing of the long-term future but only one is likely to remain, if he is lucky.
"It's a night game at Anfield, if the crowd is good then it would help", he added, with Liverpool, who lost to Manchester City in last year's final, trailing 1-0 from the first leg.
Anfield was not silent for long, though, as Firmino halved the deficit to end a insane seven-minute spell with a bullet header of his own after Martin Olsson had misjudged James Milner's cross.
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The Reds eventually leveled matters in the 69th minute when Georginio Wijnaldum's cross found Firmino, who volleyed into bottom corner.
"If we win every game from now, everybody sits here and smiles".
The only victory did come in their last league outing away from home and a similar result at Anfield would give them consecutive away league triumphs for the first time since May 2015.
- Liverpool's unbeaten Premier League run at Anfield has ended after 17 matches (W11 D6). These were there only three efforts on target.
The ease with which he scored his two goals will be of concern to Liverpool manager Juergen Klopp, whose third-place team has conceded more goals league than any of its top-seven rivals.
Asked if he would still be at Swansea after the January transfer window, Iceland worldwide Sigurdsson said: "Yeah, unless something unbelievable changes which is not down to me".
"The first time I was involved in this team was Crystal Palace and he showed all the qualities he had in that game".