The Rangers lead the best-of-seven Eastern Conference final 2-0 heading back to New York for Game 3 on Thursday and Game 4 on Sunday.
St. Louis scored a day after he and his teammates attended the funeral of his mother, who died just before Mother's Day. The Rangers have rallied around the grieving veteran and have won five in a row.
Max Pacioretty scored for Montreal, which outshot New York 41-30.
The Canadiens had Dustin Tokarski playing his first career NHL playoff game in goal in place of Price, the Olympic gold medalist who hurt his right knee in the series opener when Chris Kreider crashed into the net.
The Canadiens rebounded from a 7-2 thumping in the opener with one of their better games of the playoffs, but Lundqvist had a nearly flawless game.
Montreal's worst fears were realized in the first period, which it dominated while falling behind 2-1.
A long period of sustained pressure produced the first goal as Lundqvist's clearing attempt went off Pacioretty's skate and into the net at 6:14.
But 17 seconds later, New York tied it as McDonagh swiped a puck toward the net from the point and saw it go in off Josh Gorges' leg and the post.
The crowd was booing Kreider when he sent a pass across to the right side and Nash's one-timer went in off the midsection of Tokarski, who was late getting across at 18:58 of the first. It was the second goal in as many games and second of the playoffs for Nash.
Tokarski had no chance as St. Louis wired a shot from the slot on a power play inside the post at 8:03 of the middle period.
The Rangers played a thorough defensive game in the third to prevent a comeback.
NOTES: Montreal's Alex Galchenyuk returned from an injury to see his first playoff action for Montreal, replacing Michael Bournival. ... New York forward Derick Brassard, who suffered a suspected shoulder injury in Game 1, was replaced by Dan Carcillo. ... Price was photographed at the Bell Centre with a brace on his right knee. ... Canadian Prime Minister Stephen Harper attended the game.