Wyatt Johnston had a goal and an assist for his first NHL multipoint game, and Jamie Benn had two assists for Dallas (13-6-4), which has earned a point in eight of its past nine games (5-1-3).
"This was a playoff-type game for both teams," DeBoer said.
Scott Wedgewood, who got the start after Jake Oettinger was unable to play because of an illness, made 23 saves.
"As a backup, you've got to prepare in case of anything like this," Wedgewood said. "I took my nap, I ate my pregame meal. You've got to be ready."
Brandon Saad scored, and Binnington made 20 saves for St. Louis (11-11-0), which has lost three of four after winning its previous seven.
"It was a pretty tight-checking hockey game," Saad said. "They have a good team over there, and for the most part I thought we managed the game pretty well. But obviously, we need to create a little bit more action. Their goalie made some good saves that we could have found the back of the net with. They capitalized and we didn't tonight."
Wedgewood kept it scoreless at 13:37 of the first period when he robbed Pavel Buchnevich with a sprawling glove save.
"Sometimes you've got to be good and lucky," Wedgewood said.
Johnston then made it 1-0 at 3:51 of the second period, scoring from the top of the right circle through a screen.
"We were able to play with the lead and gain a lot of confidence," Johnston said.