Jaden Schwartz and Tyler Bozak each had a goal and an assist, and Jordan Binnington made 28 saves for the Blues, who are 4-0 on the road in the Stanley Cup Playoffs.
"Once again tonight, it's not the way we drew it up before the game, but we'll take it," Blues captain Alex Pietrangelo said.
Alexander Radulov, Andrew Cogliano and Tyler Seguin scored, and Bishop made 30 saves for the Stars.
"It was a great playoff game. It was back and forth, both teams were giving what they had," Stars coach Jim Montgomery said. "I just thought the Blues did a better job possessing pucks in the offensive zone than we did."
Cogliano tied it 2-2 at 13:06 of the third period with a shorthanded goal, knocking in a loose puck at the top of the crease after Mattias Janmark's cross-crease pass was stopped by Binnington.
Pietrangelo scored on a one-timer from along the left boards to put St. Louis ahead 3-2 at 14:24, but Seguin took a cross-ice pass from Miro Heiskanen and tapped the puck into an open net to tie it 3-3 at 15:52.
After Blues defenseman Colton Parayko was penalized for delay of game with 44 seconds remaining, Dallas pulled Bishop for a 6-on-4 advantage but had three shots blocked and didn't get one on net.
"The guys that are going out there, they have a lot of pressure on them," Blues defenseman Vince Dunn said. "But I think they really handled themselves really well."
Schwartz gave the Blues a 1-0 lead at 1:27 of the first period, redirecting Parayko's shot from the right point while tied up with Stars defenseman Taylor Fedun.
Radulov tied it 1-1 at 17:12 on the power play. Jason Spezza got the puck low along the left boards and sent a pass in the slot between Carl Gunnarsson and Alexander Steen to Radulov, who scored on a one-timer.
Bozak scored to make it 2-1 at 8:30 of the second period. Dunn skated down the right wing and fed Robert Thomas for a shot from the right face-off circle. Bishop made the initial save, but the puck trickled behind him, and Bozak knocked it into the net.
"They might not get one in the first period, they might not get one in the second period, but as the game goes on, they're going to wear you down," Pietrangelo said of the Blues' third line of Maroon, Bozak and Thomas.
