Sean Monahan opened the scoring just 1:01 into the tilt game in Calgary, but Adam Larsson tied it up soon after with a shot that dribbled through Mike Smith, who was making his first start for the Flames.
The Oilers added to their lead when Travis Hamonic turned the puck over to Yohann Auvitu, who scored, making it 2-1.
The second period opened with a goal by Jujhar Khaira, who sniped the puck over Smith's shoulder, giving the Oilers a two-goal lead.
Both teams scrummed when Mark Giordano hooked Leon Draisaitl, who responded with a bump of the captain, but no fights broke out. With play resuming four-on-four, Andrew Mangiapane took a hooking call, giving the Oilers 44 seconds to work with a four-on-three powerplay.
Smith stopped the first two shots he faced, but surrendered a goal on his second rebound to Drake Caggiula, who made it a 4-1 game for the Oilers.
Mike Smith was replaced by Jon Gillies midway through the game, as planned by Glen Gulutzan. Smith allowed four goals on 13 shots.
"I'm not going to judge or anything," Gulutzan said of Smith's performance. "We're really early. It's good to get him in there for 30 minutes and then get Jon [Gillies] some work time, same thing we did up in Edmonton.
"It's a little different for goalies because they see three practices, and you think all of our skates in the summer, it's pretty tough for them when guys are playing shinny hockey. So it's their work with the goalie coach here in the next 10 days and the shooters that we have and the things we set up that gets them ready."
Seconds after the goaltending change - the Oilers swapped out Laurent Brossoit for Nick Ellis, as well - Dougie Hamilton answered back with a rocket of a shot high from the half-boards, cutting the deficit to 4-2.
Gillies was tested in net, staring down Khaira on a shorthanded breakaway and denying him. He closed out the second period, with the Oilers continuing to lead 4-2.
The third period opened with a Giordano giveaway, leading to Kailer Yamamoto scoring just over Gillies' shoulder. TJ Brodie responded with a point shot on the powerplay, cutting the deficit to 5-3.
From there, the Flames generated a number of chances, including Dillon Dube dekeing out Ellis, only to hit the post. Monahan appeared to have scored his second of the game when a puck bounced into the net off him, but his goal was waived off.
With just under a minute to go, Jankowski scored shorthanded to cut the deficit to one, courtesy of a great pass by Dube.
"After a couple of posts I started to feel a little bit snakebitten but I knew I had to bear down on my chances," said Jankowski. "And you know, my linemates were making great plays, getting me the puck, and I knew I just had to bear down on that last one and thankfully it went in."
Gulutzan said the play of some of his players was to be expected in the first pre-season game of the year.
"Some of these young guys, it looked like in the first they were a little tentative. As they went in the third they started just to play," Gulutzan said. "When you got guys that are knocking at the door you have to give them a couple of games here in exhibition season. Let them do their thing that made them successful in the American League here and realize that you know what, yeah, it is a different league, but they're good players and once they adjust a little bit here you're going to see the real value in their play."
The Flames outshout the Oilers 33-22. Their next game is on Sept. 20, when they host the Vancouver Canucks at 7 p.m. at the Scotiabank Saddledome.
