Draisaitl tied it 2-2 on the power play at 18:03. He took a pass from Darnell Nurse at the left face-off circle and one-timed a shot past Hellebuyck.
Wheeler scored 29 seconds later to make it 3-2 at 18:32, capitalizing on a turnover by Oilers rookie defenseman Caleb Jones.
Draisaitl tied it 3-3 at 7:23 of the third period, taking a pass in front from McDavid and scoring with a quick backhand shot.
"I thought those guys that we go to every game played their hearts out," Oilers coach Ken Hitchcock said. "McDavid, Draisaitl, Nugent-Hopkins played their hearts out, and we need more support for those players. The cavalry isn't coming for those players. We need the bottom end of our team to start playing better if we expect to get to the next level. We're covering our own weight at the top, but we need better play from the bottom end of our group."
Lemieux tipped a shot from Joe Morrow at 14:02 to put the Jets up 4-3. It was the rookie's first two-goal game in the NHL in his 32nd game.
Hellebuyck made a glove save on Draisaitl with 23.2 seconds left to preserve the win.
"I had time to take a look and find out where [the Oilers] were," Hellebuyck said. "I knew [Draisaitl] must be backdoor where I couldn't see him. So when [Nugent-Hopkins] got that puck when it popped out, I just knew exactly where it was going. He was on his backhand, so he was not going to shoot that, so I knew exactly where he was going."