Burton took an early lead through Michael Simpkins before Andy Taylor pulled Exeter level after the break.
A closely-fought game looked like it was heading for a draw until Flack's late header and Devine's tap in gave the Grecians a major boost in their push for the play-offs.
Burton stunned Exeter when they went ahead in the second minute with a freak goal.
Simpkins over hit a cross from the left and the ball flew over the head of goalkeeper Paul Jones and into the top corner.
Les Afful flashed a shot across the face of goal and Edwards scooped a shot over the bar as Exeter pushed forward.
The Grecians were poor in a scrappy first half and could have few complaints about trailing at the interval.
But it took only five minutes after the restart for Exeter to pull level.
Simpkins attempted to head a throughball back to his goalkeeper but instead it landed in the path of Taylor, who made no mistake with a shot from 12 yards.
Taylor almost made it two when he connected with Alex Jeannin's cross but his powerful shot rattled against the crossbar.
Exeter were now dominant. Lee Phillips picked out defender Gary Sawyer, who forced Danny Crane to save well with a low strike.
At the other end Andy Corbett forced Jones to make a double save after a mistake from Santos Gaia, and Michael Briscoe headed wide.
But with eight minutes left on the clock Flack came off the bench and nodded home from close range with his first touch after meeting Gareth Sheldon's cross.
In the last minute Devine made sure when he tapped home after Jeannin's corner was flicked on.