"I wasn't trying to do too much, sticking with my game and relaxing and reading the game as it came to me," Hellebuyck said of his 28-save performance.
"I thought I was a calming presence for the guys out there, and it helps when they were clearing rebounds like they were. I'll enjoy this one. You don't get too many shutouts."
The overtime period was highlighted by tight goaltending.
"That's what 3-on-3 hockey does with the best players in the world out there," said Smith, who made 23 saves. "It's exciting for you guys, not so much for the goalies."
Winnipeg (5-5-0) was playing its fourth game of a five-game homestand.
"It's good to get a result," Jets captain Blake Wheeler said. "Obviously, it's been tough around here. I think the quality of our play isn't reflective of the results we're getting recently. So, it's good to finally get one."
The Jets are undefeated in regulation in their last seven games against the Oilers. Their last loss was Dec. 1, 2016.
The victory also marked Winnipeg head coach Paul Maurice's 700th career win. He's seventh all-time and third behind active NHL coaches Joel Quenneville of the Florida Panthers (893) and New York Islanders' Barry Trotz (815).
Jets defenseman Carl Dahlstrom appeared to score his first NHL goal at 8:23 of the third period, but the play was reviewed, and Scheifele was ruled offside. The goal was wiped away.
Edmonton outshot Winnipeg 7-2 in the first period.
The Jets didn't get their first shot on goal until 8:06 and finished with another 11 attempts in the period, five that missed the net and six that were blocked.
The Oilers went on a power play with 1:56 left in the opening period and had one shot that Hellebuyck turned aside.
Winnipeg went on its first power play seven minutes in the second period and got four shots on Smith.
McDavid was then called for slashing Blake Wheeler late in the second. Winnipeg managed three shots on goal.
Shots on goal after the middle frame were 15-12 in Edmonton's favor.
Winnipeg had its third power play of the game 26 seconds into the third and Smith stopped Laine's low, one-timer blast and a shot from Dmitry Kulikov.
Less than a minute after Dahlstrom's goal was disallowed, the Jets went on the power play again. Smith denied three shot attempts, including two quick ones from close in.
Oilers: At Minnesota Wild on Tuesday.
Jets: Finish a five-game homestand against Los Angeles Kings on Tuesday.
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