Schenn scored on a shot from the left circle that trickled under Lightning goalie Andrei Vasilevskiy's arm.
"I saw Nikita Kucherov] coming and the goalie cut off the angle on me," Schenn said. "I didn't see much. I don't know if it went low glove or in between his body and his arm, but I saw it trickle it and it was a big sigh of relief."
[WATCH: [All Blues vs. Lightning highlights]
The Blues (25-22-5) have won four straight and eight of their past 11 (8-2-1).
Vasilevskiy made 38 saves for the Lightning (39-11-4), who have lost consecutive games past regulation and are 2-0-2 in their past four.
"I thought we had a little bit of a slow start as in playing with the pace we wanted to," Lightning defenseman Ryan McDonagh. "You want to try an capitalize on some chances and keep the chances against us down a little bit more."
Vasilevskiy made a diving save on a rebound attempt by Schenn at 12:38 of the first period, and also stopped a wrist shot by Schenn at 10:00 of the third period before he scored his first goal in 12 games.
"He had a lot of chances tonight, I thought that line was really good," Blues coach Craig Berube said. "He finally finished it off, but since that line has been together I've really liked his game."
Binnington made a couple of big saves down the stretch, including on Ondrej Palat from the hash marks with 1:30 remaining in the third period and a backhand from Yanni Gourde with 1:01 remaining.
"I think we played much better in the third," Gourde said. "In the first two periods we were a little bit scrambled. I think we had our looks in the third, maybe the puck wasn't bouncing our way, but you have to give them credit. They played a hard game, a heavy game, and matched our intensity."
In overtime, Binnington stopped Victor Hedman on a 2-on-1 at 4:25. He is 8-1-1 with a .931 save percentage and two shutouts in 12 games (10 starts).
"This is a great way to finish off our road trip," Binnington said. "We've really come together as a team. We're playing smart hockey, disciplined hockey, and hopefully we're going to keep this going and continue to make a push."
