The Blues needed Backes' goal after failing to hold a 3-1 lead in the third period.
"The worst lead in hockey is a two-goal lead," Blues coach Ken Hitchcock said. "But it's natural; you play the score. You don't want to get caught, so you put a safety net up. It's been like that ever since I started coaching. The thing that impressed me more than anything is the way we played in overtime. Outstanding. We went into attack mode and stayed on it the whole time. I was really impressed with the way we did OT."
Dallas trailed 3-1 after two periods, but Mattias Janmark made it 3-2 at 4:35 of the third when he split the defense in the neutral zone, went in alone and beat goaltender Brian Elliott with a wrist shot. Stars captain Jamie Benn tied the game with 2:36 remaining, scoring his fifth goal of the playoffs. Benn was alone in the slot when he picked up a loose puck after a blocked shot and quickly lifted a shot over Elliott.
"We were confident," Benn said of the Stars' mindset heading into overtime. "We're going into these overtimes with our heads up. It's anyone's game and we're going out there to win them. It's unfortunate. We took a couple of penalties and they capitalized."
Troy Brouwer had a goal and an assist for St. Louis, which also got goals from Patrik Berglund and Joel Edmundson. Elliott made 31 saves, and Robby Fabbri and Kevin Shattenkirk each had two assists.
"The win was huge," Blues defenseman Jay Bouwmeester said. "We did a good job, got out feeling good about ourselves. If you win in overtime, that's good, and bringing the series back even, that's good."