Fernando Llorente scored twice in four minutes after the break to put Swansea two up before Roberto Firmino bagged a brace to draw Liverpool level.
But Gylfi Sigurdsson scored from close range with 16 minutes left to give Swansea a 3-2 win.
Michael Owen insists Jurgen Klopp's Liverpool can still catch Chelsea in the title race
10 points is a lot but it’s not over just yet
Former Liverpool man Michael Owen
Chelsea will move 10 points clear of the Reds in the Premier League table with a win over Hull tomorrow.
But Owen believes the Blues can still be caught.
“I don’t think it’s over, it’s a long way back now [though]," Owen told BT Sport.
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Express Sport rates the Liverpool players' performances from their clash with Swansea Cty at Anfield
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How did the Liverpool players get on against Swansea?
Fernando Llorente scored twice to help Swansea beat Liverpool at Anfield
“Chelsea are expected to win against Hull, with all due respect to them.
“But they’ve got to come here [to Anfield] and they’ve still got some tough games to play.
“10 points is a lot but it’s not over just yet.”