Pavel Buchnevich also scored, and Joel Hofer made 26 saves to win his fourth straight start for the Blues (15-14-3), who had lost their past two games (0-1-1), including 2-1 in overtime to the Dallas Stars on Saturday.
Blues coach Jim Montgomery said prior to the game the Blues needed to have better checking habits, especially in the defensive zone.
"Significantly better," Montgomery said. "Even in the third. They scored two goals. We hadn't given them that much. I just thought we got loose with our intensity and our checking in the third."
Brett Berard and Will Cuylle scored for the Rangers (15-14-1), who have lost four of five, including 5-1 at home to the Los Angeles Kings on Saturday. Jonathan Quick made 21 saves.
"It's frustrating," New York coach Peter Laviolette said. "It's a loss. We pressed in the third period and we didn't get it done."
New York played without leading scorer Artemi Panarin, who missed his first game this season with an upper-body injury. Forward Kaapo Kakko, the No. 2 pick in the 2019 NHL Draft, was a healthy scratch.
"We were putting fresh legs into the lineup tonight," Laviolette said of Kakko's healthy scratch. "That's based on the decision we made to go with this lineup.
"Those are the things we worked on internally through the day."
Buchnevich made it 1-0 at 18:51 of the first period on a one-timer from the bottom of the right circle off a pass from Jake Neighbours.
"I knew [Neighbours] was there, but that's a hard play for him to make," Thomas said. "He's facing the other way and he's got to spin, saucer pass. ... It's a great play by him."