Reinhart, Stephane Legault and Henrik Samuelsson opened up a 3-0 lead for the Oil Kings (19-7-5) and appeared to be cruising to victory.
But, Cody Sylvester and Alex Roach scored 26 seconds apart with 3:06 remaining in the third period to get the Hitmen (20-7-3) back in the game.
Dylan Wruck, however, added an empty-net goal in the final minute for Edmonton.
Laurent Brossoit stopped 29 shots for the win as Chris Driedger made 30 saves in a losing cause.
Edmonton went 2 for 2 on the power play while Calgary went 0 for 4.
The Oil Kings and Hitmen are tied atop the Central Division standings with 43 points apiece, but the Hitmen have a game in hand.