Zach Hyman gave Edmonton a 1-0 lead at 10:29 of the first period from the bottom of the left face-off circle after a cross-ice pass from McDavid.
Andreas Athanasiou tied it 1-1 when he took a pass from Domi and scored on a breakaway at 13:20.
Chicago went up 2-1 at 15:38 on goal by Reese Johnson, who had not played since Oct. 13. Johnson picked up the rebound of Caleb Jones' shot from the right point and backhanded it for his first goal of the season.
McDavid tied it 2-2 just 18 seconds into the second period, skating down the left side and firing a wrist shot over Stalock's left shoulder.
Jonathan Toews put the Blackhawks back ahead 3-2 at 1:33 on the power play. Toews, standing at the top of the crease, tipped in a shot from Domi at the left circle.
McDavid made it 3-3 on the power play at 8:28. His shot hit Stalock and deflected up and over him following a cross-ice pass from Draisaitl.
Domi scored to give Chicago a 4-3 lead on a 5-on-3 power play at 16:46.
There were 14 minor penalties called in the second period.
"I thought we started the game well, and then it became a lot of penalties," Oilers coach Jay Woodcroft said. "I thought our penalty-kill competed hard. Our power play got the job done and found a way to get two points on the road. We'll bank them and move on. It wasn't exactly our script. It wasn't how we wanted to win the game, but in the end, we found a result."