The game looked to be heading towards a draw before two late goals from Spanish striker Miguel Michu snatched all three points for Michael Laudrup's side.
Spanish star Michu grabbed the second with five minutes remaining of the first half before Dyer scored his second goal just after the break after being set up by Wayne Routledge.
Arsenal moved above Everton into fifth with a 1-0 win at Sunderland as Santi Cazorla scored the only goal, while Swansea went above Liverpool in seventh after beating bottom team Queens Park Rangers 4-1 thanks to two goals from Michu, who now has 15 in the EPL this season.