Kulda snapped a shot from a sharp angle six minutes into the third period that caught Medicine Hat netminder Cam Lanigan off his post.
Curtiz Lazar had a pair of goals and Mitchell Moroz scored once to round out the offence for the Oil Kings (7-3-3).
Trevor Cox and Curtis Valk had the goals for the Tigers (4-8-1), who have lost seven of their last eight games.
Cox opened up the scoring exactly one minute into the contest, finishing off a two-on-one with Valk. With the goal, the centreman extended his point streak to four games, which includes two goals and three assists.
Oil Kings' defenceman, Stephen Shmoorkoff, faced a major and game ejection at 14:45 after leaving his feet and drilling Cox with an open-ice elbow.
Valk capitalized on the ensuing power play to make it 2-0 for the Tigers.
Lazar put the Oil Kings on the board with a power-play goal at 9:24 of the second, deflecting a rocket from the point by Dylan Wruck.
With just over four minutes to go in the middle period Keegan Lowe sprung Moroz in alone, who lifted it up above the right pad of Lanigan to send it to the third tied at two.
Lazar added the empty-net insurance goal with 1:19 to close out the scoring.
Laurent Brossoit stopped 22 shots for the win as Lanigan turned aside 31 shots in defeat.
Edmonton went 1-for-4 on the power play while Medicine Hat went 1-for-5.