The best-of-7 series is tied 3-3. Game 7 is at St. Louis on Tuesday.
Jordan Binnington made 22 saves, and Alex Pietrangelo and David Perron scored for the Blues, who are 5-1 on the road in the Stanley Cup Playoffs.
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"I've said it all series, I've said it all season, that this is a group that's going to stick to that game plan," Pietrangelo said. "It might take us to the third period, (but) we're just going to keep on playing the way we need to play."
Ben Bishop allowed four goals on 20 shots for the Stars before leaving in the third period. He was replaced by Anton Khudobin, who made five saves.
"We knew this was going to be a long series, they're a good team," Stars captain Jamie Benn said. "They play good team defense and got a great goalie, so we've got to find a way to try and put some in the back of the net."
Pietrangelo scored 1:03 into the first period on a wrist shot from the point that beat Bishop blocker side through a screen to give St. Louis a 1-0 lead.
The Blues recorded the first eight shots on goal before Justin Dowling got the first for the Stars at 8:25.
Tyler Seguin tied it 1-1 at 11:35 on the power play, taking a pass in the slot from Mats Zuccarello and scoring on a one-timer that trickled past Binnington.
"We expected them to come hard and they had a good start," Seguin said. "We would have liked to have had a better start, but it doesn't matter what's happened now. It's one game and looking forward to getting on the road tomorrow."
Perron made it 2-1 at 15:24 of the second period after tapping in a cross-ice pass from Oskar Sundqvist while fighting off Stars defenseman John Klingberg.