Kassian made it 3-2 at 6:38 of the second period, and McDavid tied it 3-3 at 17:32 when he redirected a pass from Russell past Andersen.
"I thought we played a [heck] of a game once we got our wits about us," Edmonton coach Todd McLellan said. "We were turning pucks over, and it took a little regrouping between the first and the second. We didn't give up much after that, but we gave up a lot early, and that's something we have to shore up."
Nylander gave Toronto a 4-3 lead with a power-play goal at 18:29.
The Maple Leafs were 2-for-2 with the man-advantage. The Oilers were 0-for-2.
Russell tied it 4-4 at 3:01 of the third period with a one-timer from the blue line.
On the winning goal, Russell was attempting to play the puck behind the Edmonton net from the bottom of the right face-off circle after Brossoit made a save against Marleau, but he inadvertently shot it in with Kadri pressuring him.
"It was obviously an unfortunate way to end it," McDavid said. "I think everyone would have wanted overtime, it would have been entertaining. It's just the way it goes. I think we can feel good about how we battled, and no one can say we didn't put our best foot forward."