Bo Horvat scored his second goal of the game less than six minutes into overtime to lift the Canucks over the Blues 4-3 inside the NHL Edmonton bubble.
Tanner Pearson scored on the power play in the second and Elias Pettersson tallied in the third with the man advantage to give Vancouver a two-goal advantage.
Jacob Markstrom made 34 saves in the win.
Jaden Schwartz scored with seven seconds left for St. Louis in the third to force overtime.
David Perron and Alex Pietrangelo each had a pair of assists. Jordan Bennington made 21 saves in the loss.
The defending Stanley Cup champion Blues are down 0-2 in the series.