Captain Bo Horvat snapped in the game-winner 1:19 into the extra frame for his second goal of the game. He also registered an assist on the night.
Nils Hoglander also scored and notched a helper in regulation for the Canucks (16-20-3), who were playing their first game since March 24.
William Nylander had a goal and an assist for the Leafs (29-12-4), while Auston Matthews added a goal in his return from a nagging wrist injury and John Tavares contributed a pair of assists.
Jack Campbell stopped 21-of-24 shots for Toronto and Vancouver got a solid 37-save performance from Braden Holtby.
The Leafs remain atop the all-Canadian North Division and hold a 4-2 edge in the season series with three games to go.
The Canucks suffered the NHL's worst COVID-19 outbreak when the P. 1 variant of the virus swept through the team late last month.
Twenty-one players and four coaches tested positive for the virus, one other player was deemed a close contact, and another received a false positive result. Family members -- including one player's pregnant wife -- also fell ill.
No Canucks players remained on the NHL's COVID-19 protocol list Sunday. Nineteen were on the list at the height of the outbreak.
But some of Vancouver's regular starters remained out of the lineup, including netminder Thatcher Demko, defenceman Nate Schmidt, and forwards Tyler Motte and Jake Virtanen.
Toronto nearly sealed the game in the dying seconds of regulation.
Matthews dished a short pass to Mitch Marner in tight and the right-winger sailed a shot off the crossbar and post with about 20 seconds left on the clock. The goal was waved off on the ice but went to a video review.
The Leafs held a 2-1 lead going into the third period but Hoglander tied it up at 12:03.
Defenceman Jalen Chatfield sliced Hoglander a crisp pass and the rookie popped it in for his eighth of the year.
Chatfield's assist on the play was his first NHL point.
Hoglander celebrated with a big fist pump and Holtby came out of his net to join the celebratory glove taps at the Canucks' bench.
With just under five minutes to go, it briefly appeared as though Vancouver had taken the lead when forward Jayce Hawryluk rang a shot off the crossbar. He celebrated as though the puck had gone in but it was quickly waved off by the officials and the play continued.
A nasty hit 10 minutes into the middle frame saw both sides lose a key piece.
Canucks defenceman Alex Edler took out Zach Hyman in front of the Leafs' bench. The left-winger dropped hard to the ice and stayed down for several minutes before the play ended and a trainer came to his aide. Hyman went directly to the locker room and did not return to the game.
The Leafs' bench vocally expressed their displeasure with the play and it was reviewed, with officials determining Edler had committed a knee-on-knee hit. He was handed a five-minute major and a game misconduct.
Holtby kept the Canucks in the game during the ensuing penalty, at one point going from post to post to make a diving glove stop on Alex Galchenyuk.
Toronto finally beat the netminder with 19 seconds left in the man advantage. Matthews collected a pass from Nylander as he powered to the net and chipped it up over Holtby's shoulder, hitting the crossbar on the way in.
It was the star centre's league-leading 33rd goal of the season.
The Canucks responded just 86 seconds later.
Horvat streaked into the Leafs' zone and unleashed a blast from the right faceoff dot, slashing Toronto's lead to 2-1 with his 15th goal of the year.
The Leafs took an early lead Sunday following a bobbled Canucks pass in the neutral zone.
The puck ricocheted off Brock Boeser's stick and was quickly gobbled up by Tavares for an odd-man rush the other way. Tavares dishes it to Nylander, who ripped a shot past Holtby to open the scoring 3:45 into the first.
The goal was Nylander's 14th of the season and the 100th of his NHL career. The 24-year-old right-winger returned to the lineup Sunday after missing five games due to COVID-19 protocols.
The Leafs and Canucks will meet again in Vancouver on Tuesday.
NOTES: Jimmy Vesey and Travis Boyd played their first games against Toronto since being waived by the Leafs last month. ? Left-winger Tanner Pearson returned to Vancouver's lineup after suffering a leg injury on March 17. ? Before puck drop, the Canucks celebrated head equipment manager Pat O'Neill who was overseeing his 3,000th game. The team played a video with congratulatory messages from his daughters and a number of former Canucks, including Daniel and Henrik Sedin, Markus Naslund, Trevor Linden and Pavel Bure. O'Neill is the fifth trainer in league history to achieve the milestone.