Stastny took an Andrew Copp feed in the neutral zone on his off wing before moving into the Edmonton zone and unleashing a wrister through both Oiler defencemen Dimitry Kulikov and Adam Larsson which sailed over Mike Smith's glove just inside the short side post to give the Jets the win.
"Normally in the regular season you wouldn't shoot from there," said Stastny. "I think I was kind of looking for Copper (Andrew Copp). A lot of times from..if I'm gonna shoot from there I'm always trying to look for a screen." There was no scoring in the first period but the Jets outshot the Oilers by a 17-11.
Edmonton outshot Winnipeg in the second by a 12-8 margin and by one shot in the third period.
Connor Hellebuyck stopped all 38 shots he faced to earn the shutout.
Oilers captain Connor McDavid and Leon Draisaitl-who finished 1-2 in NHL scoring during the regular season-were held scoreless for the second straight game.
Oilers Head Coach Dave Tippett shifted the two back together for game two, and the pair generated more offensive chances but the pair were stymied by a constricting Winnipeg defense.
The Oilers failed to capitalize on their power play opportunities as they were given a two-man advantage for 21 seconds early in the third period but failed to score.
Mike Smith made 35 saves for the Oilers.
Games 3 and 4 are in Winnipeg on Sunday and Monday.
SHOTS-WINNIPEG 36 EDMONTON 38
NOTES-Pierre-Luc Dubois returned to the Jets lineup for game 2 while winger Nikolaj Ehlers did not. He may return on Sunday after missing the last nine games of the regular season with an injury.