Dillon Dube also scored for Calgray, and Jacob Markstrom stopped 30 shots while making his 10th straight start. The Flames pulled four points behind Montreal for the fourth and final playoff spot in the North Division.
''I guess all we can do is keep winning and hope that they lose some to make this very interesting here down the stretch,'' said Flames forward Matthew Tkachuk, who had two assists.
James Neal had the lone goal for the Oilers, and Mike Smith finished with 27 saves. Edmonton is second in the North, nine points behind first-place Toronto.
It was Edmonton's third game in four nights and fatigue looked to be a factor at times.
''We were sloppy,'' Edmonton coach Dave Tippett said. ''Sloppy with the puck and turned a lot of pucks over. Smith was really good in goal and gave us a chance. We had some chances and didn't bury anything.''
The teams meet again Saturday in Edmonton.
The Flames, desperate for points as the season winds down, were physical from the start. Matthew Tkachuk set an early tone with a strong hit on Dmitry Kulikov behind the Edmonton net.
The Oilers struggled to clear the puck out of the zone ahead of Calgary's opening goal. Tkachuk got possession and set up Lindholm for a one-timer that beat Mike Smith at 6:04 of the first period.
Neal pulled Edmonton even with his first goal since March 10. He had space down low and snapped it under Markstrom's right arm with 6:32 left in the opening period.
Markstrom stoned Leon Draisaitl late in the first to keep the game tied.
Calgary regained the lead at 7:07 of the second period after Lindholm won a faceoff, moved towards the net and deflected Rasmus Andersson's point shot.
''Obviously he's a really good two-way centerman for us but it's nice to see him get rewarded too,'' said Flames coach Darryl Sutter.
On the next shift, Neal nearly pulled the Oilers even again. He broke in alone and tried to chip it past Markstrom but the goalie made a nice glove save.
Smith outdid him at the other end in the waning seconds of the period with a big stop.
Mikael Backlund flipped a saucer pass to Lindholm on a 2-on-1 short-handed break but a sprawled, outstretched Smith managed to get his glove on the shot to deny the Calgary center a hat trick.
''It's funny how it works,'' Lindholm said. ''Two greasy ones in the beginning and then you get a really good chance and it doesn't go in.''
Connor McDavid and Draisaitl, one-two in the NHL scoring race, started to buzz more often in the third period but Markstrom was up to the task.
Calgary made it a two-goal advantage at 12:08. Milan Lucic fired a shot that rolled between Smith's legs and Dube tucked in the loose puck from a tight angle.
The Oilers pulled Smith with two minutes left in regulation but had difficulty providing sustained pressure.
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