Trevor Cox had a goal and an assist for the Tigers (21-22-3) while Elgin Pearce's second-period goal stood up as the game winner.
Keegan Low scored the lone goal for the Oil Kings (30-11-5) as Tristan Jarry turned aside 25 shots in a losing cause.
Cox opened the scoring at 14:41 of the first period on a two-man advantage as he wired a point shot off the post and in behind Jarry.
Pearce doubled the Tigers' lead with the game-winner at 11:03 of the second period when he lifted a sharp-angle shot over Jarry's glove.
Edmonton pulled back within a goal at 15:16 when Lowe's backhand pass deflected off sprawling defender Dylan Bredo and up over the pad of Lanigan.
Medicine Hat went 1 for 4 on the power play while Edmonton failed to score on three chances with the man advantage.