Lowe took a feed from Michael St. Croix from behind the net and one-timed a blast through Tigers' goaltender Cam Lanigan for the game winner with exactly one second left in overtime.
Henrik Samuelsson scored in regulation for Edmonton (16-5-4).
Dylan Bredo scored for Medicine Hat (11-15-2), which is winless in its last seven games.
The Oil Kings thought they opened up the scoring 11:38 into the first period when St. Croix scored on a wraparound. The goal was called off by officials, however, as they believed Lanigan had the puck trapped beneath him.
Samuelsson finally opened scoring at 4:45 of the second as he grabbed hold of a loose puck in front and fired home a backhand to give Edmonton the 1-0 lead.
Medicine Hat nearly tied the game six minutes into the third with Miles Koules loose at the side of the net for a one-time blast on the power play, but the rebound was cleared to safety by Lowe.
Bredo finally evened things up for the Tigers at 13:40 of the third on the man-advantage, wristing a shot from the left face-off circle over the blocker of Oil Kings' goaltender Tristan Jarry.
The goal was Edmonton's first power-play goal against in 10 games as they entered the contest with 38 straight kills.
Jarry turned aside 24 shots for the win while Lanigan stopped 45 shots in a losing cause.
Edmonton went 0-for-6 on the power play while Medicine Hat went 1-for-5.
Note to readers: CORRECTION: Updated team standings