Toronto got a good help with the return of John Tavares and Matthew Knies since John Tavares had missed six games with a lower-body injury, while Knies returned after a two-game absence.
Ryan Reaves was a healthy scratch, and it likely put much less tension after his previous suspension for a hit on Darnell Nurse in November.
Nylander hit the 30-goal mark for the fourth straight season and has become the fifth player in Leafs history to do so.
He also became just the fourth Swedish player to record five 30-goal seasons.
A key moment came late in the third period when Leon Draisaitl appeared to tie the game, but a successful coach's challenge revealed that John Klingberg was offside and it overturned the goal.
Woll then made a game-saving glove stop on Perry in the final seconds.
The Leafs continued their dominance over Edmonton by improving to 12-2-2 in their last 16 meetings and 10-2-1 in their last 13 games in Edmonton.
The Oilers haven't won a season series against Toronto since 2002-03.