Nylander's goal meant 2-2 early in the third period and came after a beautiful solo run by John Tavares.
The Swede then found the door at the front post with a quick shot.
The 24-year-old Nylander had previously been given a great opportunity to open the scoring on a free kick after just under two minutes.
He then tried to trick away goalkeeper Connor Hellebuyck who was responsible for a nice pair of legs.
Toronto had the lead with 3-2 in the final minutes when Paul Stastny steered in an equalizer puck with 2.26 left of regular time.
Matthew's hero in sudden death
The subsequent extension lasted only 59 seconds.
The Maple Leafs attacked two-on-one and Morgan Rielly hit a perfect flip-flop to the team's fixed star Auston Matthews, who shot before he split a backhand in the net.
Toronto thus won the match 4-3 and thus broke a three-match losing streak.
William Nylander is, with his eleven hits, one of three NHL Swedes who have scored as many goals this season.
The other two are Adrian Kempe (Los Angeles) and Patric Hörnqvist (Florida).
Another four Swedes are currently on ten goals scored.
Pierre Engvall was noted for an assist in the away team.