Former Manchester United midfielder Eagles held his nerve to score from the penalty spot 15 minutes from time as the Clarets ended a barren run of four consecutive npower Championship defeats.
Fabio Borini's fourth goal in five games since his loan move from Chelsea had put the visitors in front at Turf Moor early in the second half.
But the lead was short-lived as an Ashley Williams own-goal brought Burnley level and Eagles consigned the Welsh side to a fourth consecutive away defeat after Angel Rangel had fouled Jay Rodriguez.
The defeat leaves Swansea five points behind second-placed Cardiff with four games remaining, while Burnley are five points off sixth-placed Leeds with a game in hand.
Source: PA
Source: PA