The win gave the Jets a 4-0 sweep of the best-of-seven first-round series.
Mark Scheifele scored twice and Mason Appleton added a goal for the Jets, who eliminated the Oilers for the first time in history.
It took a marathon game to do it, with Connor mercifully ending the deadlock at 6:52 into the third OT.
Connor McDavid, Ryan Nugent-Hopkins, and Alex Chiasson scored for Edmonton.
Mike Smith stopped 39 shots for the Oilers and Jets goalie Connor Hellebuyck made 37 saves.
Winnipeg will face the winner of the other North Division series between Toronto and Montreal. The Maple Leafs took a 2-1 lead earlier Monday with a 2-1 victory.
The Jets will need the rest after three straight overtime games. Monday's game, the longest in Jets' history, had weary players gasping for air when they headed to the bench.
It was a shockingly swift exit for the Oilers, who had high hopes for the post-season after finishing second in the North Division ahead of third-place Winnipeg, and took the regular-season series versus the Jets 7-2.
Instead, the stunned Oilers, who haven't won a playoff series since 2017 (versus San Jose), saw their playoff dreams extinguished in just six days.
The Jets struck first on a power-play goal by Scheifele at 6:16 of the first period on a tick-tac-toe play.
Just a minute and 17 seconds later, McDavid, the NHL's leading scorer in the regular season, finally found the back of the net with his first goal of the series, scoring on a wraparound after a give-and-go with Leon Draisaitl.
The Jets regained the lead off Appleton's beautiful deflection of a Josh Morrissey shot at 15:55 of the first.
Nugent-Hopkins tied it up again at 3:44 of the second for his first points of the series, scoring with a back-hander into the top corner.
The Oilers punished the Jets on a power play for their first lead of the game at 16:37 of the second. Chiasson patiently held onto a rebound then poked the puck past Hellebuyck to make it 3-2. The normally reserved McDavid threw his hands in the air in delight.
Edmonton's joy was short-lived, as Scheifele connected for his second of the night, a slap shot at 6:01 of the third that sent the game to overtime for the third straight night.
TV cameras caught McDavid shaking his head in frustration on Edmonton's bench.
Jets' captain Blake Wheeler took a shot in the groin late in the third, and headed straight to the locker-room bent over in pain. He was back on the ice for overtime.
It was the Jets' first series win since 2018, when they dispatched Minnesota in five games, then Nashville in seven before losing to the Vegas Golden Knights in the conference finals.
Because of COVID-19, the game was held without fans at Bell MTS Centre, an unfortunate fact Jets players lamented after the previous night's thrilling win. But while no spectators were permitted inside, an endless line of cars, flags flying, horns honking, paraded down Portage Ave., past the arena before the game.
Only a night earlier, the Jets completed one of the craziest comebacks in NHL history.
Sparked by Josh Archibald's tripping penalty on Logan Stanley, Mathieu Perreault scored on the ensuing power play, and opened the floodgates. The Jets scored three goals in three minutes and three seconds, and clinched the win in overtime.
Winnipeg became just the 10th team in NHL history to come back from three goals down in the final 10 minutes of the third period in the post-season.
Archibald was given a one-game suspension and missed Monday's game.
This was the first post-season series between the two teams since 1990, when Edmonton ousted the Jets in seven games in the Smythe Division semifinals.
Winnipeg won Games 1 and 2 in Edmonton by scores of 4-1 and 1-0.