Wheeler scored on the first shot of the tiebreaker, causing goalie Viktor Fasth to stretch out and injure himself. He left the ice and was replaced by Richard Bachman.
Michael Hutchinson stopped 18 shots in the victory and has won all four shootouts he's been in this season.
Fasth made 41 saves through regulation and overtime.
Bryan Little scored a pair of goals and added an assist for Winnipeg, which improved its record to 30-19-10 and 4-1-0 in its past five games. Andrew Ladd also had a goal and defenceman Tyler Myers scored his first as a Jet after being acquired in a trade last week with Buffalo. Toby Enstrom and Wheeler each had a pair of assists.
Edmonton (16-32-10) got a pair of goals from Nail Yakupov and Anton Lander. Ryan Nugent-Hopkins also scored as the Oilers finished their six-game road trip 2-3-1.
Nugent-Hopkins gave the Oilers a 3-2 lead at 1:08 of the third period with his 15th of the season, but Myers evened the score at 3:30.
Yakupov netted his eighth of the season at 15:14 on the power play, but Ladd responded with his 20th just 42 seconds later to make it 4-4.
Winnipeg outshot the Oilers 46-2 through regulation and overtime.
The Jets got some unintentional extra help in scoring the game's first goal.
With Edmonton defenceman Keith Aulie in the penalty box for tripping, the Jets went on the power play. About halfway through it, the stick of Oilers forward Benoit Pouliot broke in half. Pouliot tried to use his body as the Jets swarmed around Edmonton's net, but Little flipped a rebound past Fasth at 7:46 for his 22nd goal of the season.
Jets centre Mathieu Perreault limped off the ice early in the first period after a hit to the boards and didn't return.
Edmonton took a 2-1 lead when it scored a pair of goals 90 seconds apart, first a low shot from Yakupov at 13:49 and then a power-play goal from Lander off a rebound at 15:19 while Ladd was in the box for hooking.
Yakupov extended his points streak to five games, including three goals and three assists.
Winnipeg was ahead 16-6 in shots on goal.
Little fired in his second of the game through traffic at 8:16 of the second period to tie the game, 2-2.
Edmonton winger Luke Gazdic left the game in the second period when Jets defenceman Dustin Byfuglien checked him hard on his left shoulder and his face hit the boards.
Jets forward Drew Stafford, also acquired in the trade with Buffalo, left the game in the third with an upper-body injury.
Winnipeg starts a two-game road trip Thursday in Washington. The Oilers begin a three-game homestand against Boston on Wednesday.