Stastny's screened wrist shot beat goaltender Mike Smith high to the right corner.
''I figured I'd try to use the D as a screen and just put the puck on net,'' Stastny said. ''I feel like in overtime, whether you're trying to score or whether you're trying to generate offense, sometimes that's the easiest way to do it.
''The puck had eyes and, fortunately, it went in.''
Connor Hellebuyck made 38 saves for Winnipeg and has stopped 70 of 71 shots over the first two games.
''He's the reason we have a chance,'' Jets captain Blake Wheeler said. ''He's the reason we are a playoff team and the reason we have confidence every time we go out there. You just can't say enough about having a guy who you believe is going to stop every shot.''
Smith made 35 saves for Edmonton, which finished second in the North Division - ahead of third-place Winnipeg - and took the season series 7-2.
Winnipeg won the opener 4-1 on Wednesday night. The series shifts to Winnipeg for games Sunday and Monday nights.
''We dug ourselves a bit of a hole,'' Oilers captain Connor McDavid said. ''Both games could have gone either way. It's not the position we want to be in. We've been a good road team all year long and had success in that building as well and we've been good on back-to-backs.''
McDavid and teammate Leon Draisaitl - the NHL's top two scorers - have been held off the scoresheet in the first two game.
''Our desperation has to get up,'' McDavid said. ''We've got to be hungry around the net. Some loose pucks laying around there that are waiting to be put in the net, we've got to send them on their way. Maybe a little hesitation in certain spots.''
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