Tyler Pitlick and Alexey Toropchenko scored, and Jordan Binnington made 22 saves for St. Louis (37-35-7), which is 8-2-2 in its past 12 games.
"We're going into games trying to win, play our game and kind of build on that for next year," Blues forward Kasperi Kapanen said. "Obviously, you don't know about next year and what the team's going to look like and whatnot. I think you want to end on a high note, you want to play well until the end and that's what we're trying to do right now."
Kapanen scored the game-winner 1:16 into overtime with a one-timer from the left face-off circle off a pass by Brayden Schenn.
"Of course, whenever you score in overtime, it feels good," Kapanen said. "'Schenner' made a great play, and I was able to put it in. ... I think we go against a forward (Vincent Trocheck) there, so I think he realized he could make a good play there and he did. Lucky it went in."
Toropchenko gave the Blues a 1-0 lead at 8:00 of the second period when he swept the puck away from Artemi Panarin with his leg in the defensive zone, skated down the left side and scored off the right post.
Tarasenko tied it 1-1 at 1:55 of the third period on a power play. He scored on a loose puck in the slot after a centering pass by Mika Zibanejad, which was intended for Chris Kreider, deflected off the skate of Colton Parayko.
"He does score a lot of goals," said Parayko, who played eight seasons with Tarasenko. "Unfortunate, but I guess cool for him to score back here at Enterprise. Obviously, you don't want him to score. It's over, I guess that's what we could say about it. Cool, I guess."