The Jets lead the best-of-7 series with Game 3 at Winnipeg on Sunday (7:30 p.m. ET; NBCSN, CBC, SN, TVAS).
Stastny scored from the top of the right face-off circle after taking a pass from Andrew Copp just inside the Oilers blue line.
"The play started in the D-zone and the puck was in the corner," Stastny said. "And [Tucker Poolman] chipped it past their guy and so then I just started picking up speed knowing we'd have a 2-on-2 or 3-on-3. Once [Copp] middle-lane drove and kicked it out to me, at first I was going to kind of look for him, but once he slashed across, I figure I would try to use the [defenseman] as a screen and just try to put the puck on net.
"I feel like in overtime, whether you're trying to score or you're trying to generate offense, sometimes that's the easiest way to do it, and the puck had eyes and fortunately it went it."
Connor Hellebuyck made 38 saves for the Jets, the No. 3 seed in the Scotia North Division. Mike Smith made 35 saves for the Oilers, the No. 2 seed.
"It's fun to be a part of a game like that, that's what you want, obviously a tight-checking game, 0-0," Oilers captain Connor McDavid said. "I thought we had some good chances. We have to find a way to bear down and get one."
McDavid, who led the NHL with 105 points (33 goals, 72 assists) in the regular season, and Leon Draisaitl, who was second with 84 points (31 goals, 53 assists), each was held without a point for a second consecutive game.
McDavid went two games without a point once during the regular season, in three straight losses to the Toronto Maple Leafs (Feb. 27-March 3).
Draisaitl went two games without a point three times (Jan 16-18 against Montreal Canadiens, Feb. 27-March 1 against Toronto, April 8-10 against Ottawa Senators and Calgary Flames).
"There is a cost to it, we're not generating a tremendous amount of offense in the game," Jets coach Paul Maurice said. "That's the price that has to be paid to be on the right side [defensively] for the most part, and if you're not, to make sure the puck stays on the outside. We've had to give a little up to give ourselves a chance -- those guys are so dangerous and so elite -- and when we haven't, then Connor Hellebuyck has been."
It was the first time the Jets went into a playoff overtime at 0-0. It was the third time for the Oilers.
"I think it's been a group effort [defensively], we have to play as a group of five and our defensemen are doing a really good job," Jets captain Blake Wheeler said. "We're trying to come back and give them help and let our goalie see the puck. It's a good job by everyone back there and [Hellebuyck] made some good saves tonight."
