With Zane Jones serving a goaltender interference penalty, Inglis roofed a rebound past Calgary goalie Chris Driedger at 2:31 of the extra period.
Tyson Ness also scored for the Rebels (1-3-0), who halted their losing streak at three games.
Brady Brassart and Cody Sylvester scored power-play goals for the Hitmen (3-0-1), who jumped out to a 2-0 lead through one period of play.
Bolton Pouliot stopped 37-of-39 shots he faced to pick up the win for Red Deer as Driedger finished with 23 saves.
The Rebels went 1-for-2 on the power play while the Hitmen went 2-for-4.