Gabe Perreault and J.T. Miller scored in the third period to give the Rangers a two-goal lead, and Vladislav Gavrikov had a goal and an assist.
The Rangers (3-1-0) have won their first two games at home this season after losing their first seven home games last season.
They didn't get their second win at home last season until Nov. 24. Their first came on Nov. 10.
"It feels so good right now to play well in front of the fans," Miller said.
Nick Schmaltz and Andrew Peeke scored, and Karel Vejmelka made 18 saves for the Mammoth (2-1-0). They were playing their second game on a back-to-back after winning in Columbus on Saturday.
"Through two periods we were probably doubling that team in scoring chances; that's a lot of good," Mammoth coach Andre Tourigny said. "But that said, that's why you play three periods and I just think mentally we did not sustain in the third. We didn't stay patient. We forced it. I think there's a lot of will and competitiveness in our team and sometimes if you don't use it the right way you force the game, you open up the play and sometimes they capitalize."
Perreault gave New York a 2-1 lead at 3:15 of the third period, when he scored off a long rebound.
Mika Zibanejad and Alexis Lafreniere came in on a 2-on-1. Zibanejad moved the puck across right to left to Lafreniere in the left face-off circle. Vejmelka came across to stop Lafreniere's shot, but the puck came out to Perreault above the left hash marks and he shot it in.
"I had the whole net," Perreault said.
Miller made it 3-1 at 12:47, when he got to the net and Gavrikov's shot from the point redirected in off his skates.
"I figure if you do it 10 times a game something will go off your foot," Miller said of going to the net. "That's the way to beat teams. When you want to play later those are the goals you're scoring, you're not getting the pretty ones. It's always been a part of my game. It's nice to see one go in."