Dylan Holloway and Brayden Schenn each scored for the Blues (7-10-6), who had a three-game point streak come to an end. Joel Hofer made 17 saves.
"I didn't think we played with the requisite of intensity and connectedness and puck support that we've been playing with the three previous games," Blues coach Jim Montgomery said.
Holloway gave the Blues a 1-0 lead at 17:36 of the first period.
He stripped the puck from Taylor Raddysh behind the Rangers net and set up in the left face-off circle. Philip Broberg got the puck back to Holloway, who scored with a wrist shot through Raddysh's legs and over Shesterkin's glove into the top right corner.
"I just stayed in a little bit of soft ice and 'Broby' made a great read, passed it over and I was lucky enough for it to go in," Holloway said.
But the Blues could not build on the lead. Instead, Trocheck tied it 1-1 at 10:06 of the second period.
Fox sent a stretch pass out of New York's defensive zone and up to Jonny Brodzinski, who split St. Louis defensemen Logan Mailloux and Cam Fowler and used his backhand to dish it to Trocheck in the right circle. Trocheck scored with a near-side shot past Hofer.
"That play there (Brodzinski) has a lot of speed, we're able to get above them in the neutral zone and then he just finds me backdoor," Trocheck said. "Great play."
Trocheck said he knew that Fowler and Mailloux were caught on the ice for an extended shift and the Rangers needed to attack them.
Fowler's shift length ended at 2:20 and Mailloux's 2:19.
"You're just trying to get it up as quick as you can without letting them change," Trocheck said. "Jonny did a great job of flying to the middle of the ice where there was a lot of room, challenged the guy one-on-one and I was able to slide backdoor."
The Rangers scored again just 40 seconds into the third period, when Lafreniere, from in front of the net with position on Broberg, got a piece of Gavrikov's floater from the left point, deflecting it down and past Hofer for a 2-1 lead.
Edstrom made it 3-1 at 8:56, scoring from in front off a pass from Gavrikov, who snuck down below the goal line as Fox was shooting from the point. Fowler and Nathan Walker left the slot to defend Gavrikov, leaving Edstrom alone in front.
"I thought we did a good job of getting inside in the offensive zone, just getting to the blue paint," Rangers coach Mike Sullivan said. " 'Eddy's' goal and 'Laf's' goal, just getting to the blue paint. That's where the majority of the goals are scored in this League. It's no secret."