Johnny Gaudreau had a big night creating plenty of offence and scored the lone marker for the visitors.
Jacob Markstrom - making his 11th straight start for Calgary - was excellent, especially in the first period when the Oilers came out with a hard push and poured pucks on net. He stopped 27 shots.
At the other end of the ice, Mike Smith made 29 saves for the Oilers.
With the loss, Calgary finished 4-6 in the season series after winning the last two tilts against Edmonton by an 8-1 total, the most recent a 3-1 victory Thursday night in Edmonton.
There were a pair of back-to-back nail-biting moments for the Flames early in the contest, beginning with Kailer Yamamoto's shot hitting Chris Tanev and the ensuing deflection nearly beating Markstrom.
Then the rebound was smacked through the crease behind Markstrom and looked like it may go off the far post and into the net but Mark Giordano was there to grab the puck and get it out of harm's way.
Markstrom was then tested again and made a massive backhand stop on Jesse Puljujarvi after the Oilers forward split Juuso Valimaki and Michael Stone between the faceoff dots.
Valimaki was called for a hook on the play and the Oilers powerplay - ranked third in the league coming into the night - capitalized at 4:46.
Connor McDavid - the league's scoring leader who had been held without a point by the Flames in the their last two outings against the Oilers - scored his 29th of the season, playing in his 400th career game.
Defending Hart Trophy winner Leon Draisaitl - who had also been blanked in the previous two tilts with Calgary - got an assist.
Smith absolutely robbed Andrew Mangiapane in tight and the Oilers went the other way, McDavid feeding Ethan Bear down low for a one-time blast that found net at 7:19. Draisaitl had the other helper.
The Flames got their first powerplay at 9:22 and while they had lots of zone time and blasts from the point, they weren't able to get any pucks on net.
They got a do-over less than a minute after the minor expired when Darnell Nurse was sent off for boarding Rasmus Andersson.
The visitors generated a pair of shots - and Mikael Backlund put one over the net on a set up from Andrew Mangiapane - but they weren't able to get on the board.
Gaudreau - who came into the game with a six-game scoring streak on the go (2+5), and had points in eight of his last nine games (4+7) - got within inches of beating Smith, firing a shot for the top far-side corner, one of his favourite spots to paint, but it clanged off the cross-bar.
Stone made a stellar, jaw-dropping stop on the Oilers second powerplay, McDavid staring at an open net when a rebound came right out to his stick, the Calgary defenceman dropping into the butterfly and getting the toe of his right skate on the puck to send it wide. Edmonton tallied six shots on the man-up.
Gaudreau scored at 10:48 off a quick up-ice play. Chris Tanev sprung Matthew Tkachuk with a long pass and he drove the net with hard wrister that Smith kicked out with his right pad, but No. 13 put it home with a sweeping backhand as he swooped in for his 18th of the season.
It was also his fifth goal in 10 games against the Oil this year.
