St. Louis is six points ahead of the Dallas Stars for third place in the Central Division.
"It's nice to get back in the win column, yeah, and just continue to do what we've been doing, take it one game at a time and keep focusing," said Binnington, who got his 14th win in 17 starts.
Charlie Coyle scored a shootout goal in his Bruins debut. Coyle, a forward, was traded by the Minnesota Wild to Boston on Wednesday for forward Ryan Donato and a conditional fifth-round pick in the 2019 NHL Draft.
"I had my linemates kind of help me out and talk to me a lot, which is huge," Coyle said. "I think I adjusted decently. I'll learn as we go along here."
Sammy Blais scored in the sixth round of the shootout, and Tyler Bozak also scored for the Blues.
Blais, recalled from San Antonio of the American Hockey League on Thursday, did not expect to be called upon.
"I thought they were going to choose someone else, but it worked out, so I'm pretty happy about that," Blais said. "When [Rask] went down a little bit, I saw the opening and I just shot it, so it worked."
Steen's first goal since Nov. 30 (27 games) gave the Blues a 1-0 lead at 7:15 of the first period.
Wagner tied it 1-1 for the Bruins at 5:12 of the second.