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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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.”