Vancouver took the lead early in the second period when Elias Pettersson found JT Miller but a pass across the ice.
Then St. Louis thanks to goals from first Justin Faulk and then David Perron.
But despite Perron's full hit coming with less than two minutes left of the second period, Pettersson had time to equalize before the team went to the break.
Just over half a minute after the game started again, the Blues defenders were careless and Pettersson got hold of the puck and shot it coldly behind Jake Allen.
The score was 2-2 all the way to extra time and it also took a quarter of an hour before Brayden Schenn came free with Markström and decided the match to 3-2.
Vancouver now leads 2-1 in matches. The next match will be played early on Tuesday morning, Swedish time. The entire match series is decided in Edmonton.
CLIP: Three great Swedish goals in the NHL
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Elias Pettersson in full swing during the Stanley Cup play-offs Photo: Bildbyrån