Martin Rucinsky had a goal and an assist for the Rangers, who scored three goals on their first 10 shots against Theodore, who left after the first period.
"Obviously he's an incredible goalie," said Moore, who also beat Canadiens rookie Yann Danis for the Rangers' final goal. "I don't think that's something we were focused on or patted ourselves on the back about. We were just focused on the things that make us successful."
"They came on strong, but Henrik really had a calming effect for us," said New York center Steve Rucchin, who scored New York's fourth goal midway through the third. "He's pretty levelheaded back there and we knew he was going to keep making the big saves."
Lundqvist stopped 38 shots -- including 17 of 19 in the third -- as New York won its second in a row following a three-game winless streak.
"They kept coming so hard, you couldn't relax," Lundqvist said. "It was really nice when we scored the fourth and fifth goals. It was a huge win for us."
Jagr extended his points streak to nine games when he beat Theodore midway through the first period after Moore opened the scoring 2:46 in.
Danis, who had a shutout in his NHL debut on Oct. 12 in Atlanta, made his first Bell Centre appearance to begin the second.
Montreal, which lost its second in a row, got goals from Alex Kovalev and Richard Zednik early in the third to draw within 3-2.
"You can lose it right away," Jagr said when asked about the difficulty of protecting a lead in the new NHL style, resulting from rule changes this season. "That's why it's exciting. It's never over. You can be down four goals and easily win the game."
Danis, who stopped all 10 shots he faced in the second, saw his shutout streak end at 89 minutes, 34 seconds when Rucchin scored on New York's second shot of the third period.
"We had the momentum and things were really going well at that point for us, and obviously it's disappointing to give up that goal, but I wasn't thinking at all about the shutout streak," Danis said.
Moore restored the Rangers' three-goal lead moments later on New York's next shot, beating Danis high at 10:12.
The Canadiens ended a three-game home winning streak, falling to 7-3-1 overall.
"They came out hard," said Montreal captain Saku Koivu, who assisted on both Canadiens goals. "The first 10 minutes were really strong for them and we just didn't show up."
Moore opened the scoring with a nice individual effort. The Rangers center swept down the right side and went around defenseman Andrei Markov, cutting behind him and across the goal mouth to tuck the puck past Theodore.
"I had a head of steam from our end, and once I sort of got the corner, got positioning on him, it's tough for him to recover," Moore said.
Jagr made it 2-0 at 10:37 when his flubbed shot floated in and struck Theodore's glove before popping out behind the Canadiens goalie and into the net.
Theodore had little chance on the Rangers' third goal at 11:45. The former Hart Trophy winner stopped Marek Malik's point shot but kicked the rebound directly to Rucinsky, who fired a quick, sharp-angled shot from the corner into the net to make it 3-0.
Kovalev, who hit the left post late in the second, scored his third of the year 51 seconds into the third to ruin Lundqvist's shutout bid.
Zednik brought the sellout crowd of 21,273 out of their seats again at 4:28 when he scored his first of the season to draw the Canadiens within 3-2.
- Moore had one point -- a goal against Montreal on Oct. 6 -- in his previous 11 games this season. His six points in 17 career games have all come against the Canadiens.
- Lundqvist improved to 5-1-2 in eight starts.
- Rucinsky left in the third period with an apparent knee injury after taking a hard, late check from D Craig Rivet along the side boards.
- The Canadiens have sold out each of their six home games.