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 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.
Eagles had an early chance to put the Clarets in front but he fired wide from the edge of the penalty area.
Swansea started to take control and when leading scorer Scott Sinclair cut inside and shaped up to shoot from just outside the penalty area the Clarets must have feared the worst, but his left-foot strike was off target.
Midfielder Darren Pratley almost marked his return to the Swans' starting line-up with a goal 17 minutes in but his effort from 12 yards struck the post.
Andre Bikey spurned an opportunity to put Burnley in front against the run of play when he headed wide.
But the visitors were denied by the woodwork once again when Pratley picked out Sinclair and the former Chelsea winger's header came back off the post.
Borini was wasteful eight minutes before the break when he fired wide of Brian Jensen's goal as Swansea failed to reward their dominance with a half-time lead.
But Borini made no mistake eight minutes after the break when he broke the offside trap and raced onto Pratley's through ball then kept his composure to slot the ball past Jensen.
Swansea were caught cold two minutes later though, when Eagles' strike flicked off Williams and deceived keeper Dorus De Vries to peg the Welsh side back.
Chris Iwelumo had a chance to put Burnley in front in the 66th minute when he rose to meet Daniel Fox's free-kick but the big striker headed wide from just six yards out.
Swans boss Brendan Rodgers brought on strike duo Luke Moore and Tamas Priskin as the visitors went in search of a precious victory.
But they were stunned when Burnley snatched all three points from the penalty spot 15 minutes from time.
Rangel was adjudged to have fouled Rodriguez and Eagles stepped up to beat De Vries from 12 yards out.