Dach gave Montreal a 1-0 lead at 4:54 of the second period, scoring on a 3-on-2 rush. After Jacob Trouba's shot on a short-handed 2-on-1 went wide, Caufield moved the puck to Christian Dvorak, who gave it to Dach for a shot glove side from the right hash marks.
Dach has 27 points (eight goals, 19 assists) this season, an NHL career high.
"Both games (against the Islanders and Rangers) we played pretty good goalies, and you know it's not going to be a shootout, you're going to have to end up scoring one or two and playing good defensively in a good team-structured game," Dach said. "That's what it came down to, and it's nice to be rewarded."
Panarin tied it 1-1 at 16:03. Although the Rangers failed to convert on a power play that started at 13:08, they were able to hold the puck in the offensive zone, and with Shesterkin off the ice for an extra skater because of a delayed penalty, Panarin scored with a wrist shot from the right face-off circle.
However, Caufield made it 2-1 at 8:56 of the third period, burying a shot by Shesterkin after K'Andre Miller lost the puck in his skates below the Rangers' goal line.
"I'm always ready in those situations," Caufield said. "You've got to capitalize on their mistakes, and we did that there."
It was Caufield's 26th goal on the season and his seventh in his past eight games.
"I felt from start to finish we were very engaged physically, mentally," St. Louis said. "It's one of those games where you're coming into this building, they're 13-2-2 in their last 17, they've been waiting for you. The game is at 5, a quick turnaround, we're on a back to back, our injuries, it seemed like before the puck dropped you have an uphill battle.
"I just felt we came in tonight and we didn't focus on all of the excuses we had to not give it our all. Just a gutsy win. By no means was it perfect, but the engagement was an all-time high for our team this season."
NOTES: The Canadiens won on the road for the first time since Dec. 19 (0-6-1 in previous seven). … Miller had an assist on Panarin's goal to extend his point streak to six games (three goals, five assists). … Jake Leschyshyn made his season debut with the Rangers, who claimed the forward off waivers from the Vegas Golden Knights on Wednesday. Leschyshyn had two shots and two hits in 6:47 of ice time. … New York played its second straight game without forward Chris Kreider, who is day to day with an upper-body injury.