The Hitmen now lead the best-of-seven Eastern Conference final 2-1. Game 4 is at Calgary's Scotiabank Saddledome on Wednesday.
Greg Chase, Brady Brassart, Elliott Peterson, Calder Brooks and Victor Rask scored for Calgary.
The Oil Kings' Trevor Cheek opened the scoring on the power play at 6:29 of the first period. Curtis Lazar scored Edmonton's other goal, also during a man advantage, in the third.
Chase tied the game at one for Calgary just 24 seconds into the second, then Brassard and Peterson scored within 40 seconds of each other for a 3-1 lead.
Chris Driedger made 32 saves for the Hitmen.
Laurent Brossoit stopped 22-of-27 shots for the Oil Kings before being replaced by Tristan Jarry for the third period.
The Hitmen were 2-for-8 on the power play while the Oil Kings were 2-for-5.