Calgary salvaged the victory after squandering a 3-2 lead in the final minute of regulation.
Twenty-three seconds into overtime, Gaudreau scored with a shot from low in the left-hand circle. There were some tense moments as officials reviewed the play for potential offside, but the goal was upheld after video review.
Gaudreau wasn't too worried as officials reviewed his goal.
''I looked at the replay, it looked pretty good. We were talking about it on the bench,'' Guadreau said. ''Sometimes it goes either way and it went our way tonight.''
Dillon Dube, Milan Lucic and Elias Lindholm scored for the Flames in regulation. Rasmus Andersson had two assists.
Tanner Pearson, Bo Horvat and Brock Boeser scored for Vancouver.
Jacob Markstrom stopped 30 of 33 shots for the Flames. Thatcher Demko had 23 saves for Vancouver.
Vancouver held a two-goal lead near the end of the first period but saw the game start to slip away on a series of costly mistakes.
The game was the third of a four-game series between the teams. Calgary has two wins heading into the final matchup on Wednesday.
Boeser forced overtime by scoring on shot past Demko with 30.2 seconds left in regulation. It was a short-handed goal as Vancouver defenseman Quinn Hughes sat in the box. The Canucks pulled Demko in a desperate bid to salvage the game.
''It just shows that we have no quit. Even when we're down a guy and pull the goalie and get one and get that extra point. It's good for our team,'' said Boeser, who also had an assist on Monday.
''Obviously we make a few mistakes and if we don't make those, I think we win the hockey game, so I think overall we still played a good game but some small details we've got to fix.''
NOTES: Connor Mackey registered an assist on Lucic's goal, marking the first point of the 24-year-old defenseman's NHL career. . Calgary's Mikael Backlund did not play after suffering a lower-body injury on Saturday.
The Flames and Canucks wrap up their four-game set on Wednesday.
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