The Flames jumped out to a 2-0 lead with two goals in the first 51 seconds.
Elias Lindholm beat Mike Smith with a long wrist shot, then Andrew Mangiapane put a shot from close range in off the post. Brett Ritchie scored off a turnover 6:05 into the game, chasing Smith from the game after allowing three goals on 10 shots.
Connor McDavid slid a puck under Jacob Markstrom, but the Oilers were badly outplayed in the first.
The Flames were up 3-1 after the first period and had a 19-7 advantage in shots.
The onslaught continued in the second with Blake Coleman flipping in a rebound to make it 4-1 just 45 seconds into the session.
Coleman made it 5-1 before Evan Bouchard finished off a feed from McDavid.
It became 6-2 when Matthew Tkachuk batted a puck out of the air on the power play. Zach Hyman fooled Markstrom from a sharp angle as the teams combined for nine goals before the game was 30 minutes old.
Hyman scored again with 5:54 left in the second, then Leon Draisaitl struck on a partial breakaway in the final minute to pull Edmonton within a goal.
Kailer Yamamoto stuffed in a rebound early in the second, but Rasmus Andersson came right back for the Flames to make it 7-6 Calgary.
Tkachuk took advantage of an Oilers turnover to score on a breakaway with 11:05 to go. He added an empty netter to complete the hat trick.
McDavid finished with four points. Koskinen stopped 32 of 37 shots.
Game two is Friday (630 CHED, Face-off Show begins at 6 p.m., the game starts at 8:30 p.m.)
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