Jamie Oleksiak scored for the Stars (35-20-6), who had a six-game point streak end (5-0-1). Ben Bishop made nine saves on 13 shots in 24:45 before being replaced by Anton Khudobin, who made nine saves in relief.
"We're a lot better of a team than we showed tonight, and we know that," Stars coach Rick Bowness said. "We had a bad night and it's the whole room. We're not going to point our fingers at one or two guys. It's the whole room and the coaches."
Alexander Steen scored on a second effort for a 1-0 lead at 5:48 of the first period after his first shot was blocked by Oleksiak.
David Perron scored on a rebound in front at 17:18 to give the Blues a 2-0 lead.
"It was a good win," Blues coach Craig Berube said. "We knew what was on the line."
St. Louis did not allow a shot on goal for 10:39 in the first period and outshot Dallas 10-6.
"I think it's a good learning point that we have work to do," Stars forward Jamie Benn said. "There's a lot of areas of our game that we need to keep getting better at. They're the best. Right now, we're not as good as them, so we have work to do."
Kyrou extended the lead to 3-0 at 2:24 of the second period, and Schwartz scored on a wrist shot from the slot at 4:45 to make it 4-0. It was Kyrou's first goal in 17 games.
"I thought he was good," Berube said. "He made a nice play on the first goal, for sure. And then, scoring that second goal. He's hanging onto the puck and battling for it.
"It's good to see him have a good game in this type of atmosphere tonight."
The Blues did not allow a shot on goal for a span of 10:06 in the second period.
"It was a tough game from top to bottom," Stars forward Blake Comeau said. "We didn't execute, and I felt like we were just throwing it all over the ice and were kind of spread out. When you play a team like St. Louis that is very structured and plays a hard and physical game, that's not going to work.
"We can't get too down on ourselves but we have to figure out what we did wrong and correct it."
Parayko scored on a one-timer at 2:27 of the third period to make it 5-0. Four of his seven goals this season have come against the Stars.
"The way the guys in front of me and [Binnington] have played the last four games has been pretty remarkable," Allen said. "Scoring chances created, back-checking and blocked shots. It's great to see."
Oleksiak scored from the point at 16:27 for the 5-1 final.
