Bolduc's first goal at 4:30 of the second period gave St. Louis a 2-1 lead. He found the puck loose in the slot with Shesterkin reaching for it and put it in.
Cuylle's second goal made it 2-2 at 7:20. Hofer lost his stick on Berard's wraparound, and the puck kicked off of it and went into the right circle to Mika Zibanejad, who tapped a pass to Cuylle cutting through the left circle. He scored before jumping into the cage.
The Rangers generated zone time and some chances late in the second period, which ended with a skirmish in the corner of St. Louis' defensive zone and Shesterkin and Hofer nearly squaring off at center ice.
The third period started 4-on-4 with Cuylle and Jake Neighbours both in the box for unsportsmanlike conduct penalties at the end of the second.
The Blues took advantage of the extra room. Schenn scored a breakaway goal at 1:12 to give them a 3-2 lead.
Three of the four Rangers players on the ice, including defenseman Zac Jones, were low in the offensive zone when the puck came out to Scott Perunovich on the right side. He found Schenn flying the zone behind the Rangers' defense.
"We've got a lot of guys with really good offensive hockey sense and if we are committed to checking like we were the offense is going to come," Montgomery said. "We were really connected."
The Rangers did not have much pushback, generating seven shots on goal in the third period.
"We need more of a response because it's not good enough," Laviolette said. "It wasn't good enough tonight. When something like that happens there's got to be a difference in the way we play the game, the attack we go on. That was missing."
Bolduc scored at 8:25 to make it 4-2. He found the puck between the circles off a rebound from a blocked shot and scored with a shot into the top right corner.
"We have to stop it somehow," Rangers forward Artemi Panarin said. "We're professionals. Everyone has that experience when things are not going well in your days. We've got to wake up tomorrow, fix our mistakes and be ready for Carolina [on Wednesday]."
Pavel Buchnevich's empty-net goal at 17:15 made the 5-2 final.
"There's always life and energy when any team makes an in-season coaching change and tonight it showed," Schenn said. "Now it's our job to keep on building with that."
NOTES: The Blues won at Madison Square Garden for the first time since March 3, 2020. … The Rangers lost for the first time this season when scoring the first goal (10-1-0). … Cuylle, who had his first multi-goal NHL game, leads New York with nine even-strength goals.