Stromgodset had no room for slip-ups as title rivals Rosenborg had won with a late goal of their own against Start on Saturday to take a three-point lead at the top of the table.
In a see-saw game, Stromgodset had plenty of chances and Oyvind Storflor hit the crossbar with a free kick on the hour mark, but it was not until the 82nd minute that they managed to fashion a controversial winner.
Defender Jarl Andre Storbaek appeared to handle the ball before firing it across the goal where veteran Hungarian striker Kovacs stabbed home from close range.
The win put Stromgodset back on top of the Tippeligaen on 56 points, ahead of Rosenborg on goal difference. Haugesund are in third place on 45 points with three games left to play.
The next round of games sees Rosenborg face a tricky trip to defending champions Molde and Stromgodset host Start.