Gabe Perreault, J.T. Miller and Eeli Tolvanen each scored in the third period for the Rangers (3-1-0), who have won their first two games at home this season after losing their first seven home games last season.
Shesterkin, who scored a goal in the home opener Thursday, had an assist on Tolvanen’s empty-net goal.
Nick Schmaltz and Andrew Peeke scored, and Karel Vejmelka made 18 saves for the Mammoth (2-1-0). They lost for the first time this season.
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.
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.
Utah got one goal back at 14:57, when Peeke scored with a quick shot over Shesterkin's glove from above the right face-off circle to make it 3-2.
Tolvanen scored into the empty net to make it 4-2 at 18:45.
Gavrikov gave the Rangers a 1-0 lead at 17:18 of the first period.
Pavel Dorofeyev's forecheck disrupted Anders Lee's pass out of the zone. The puck redirected to Adam Fox at the right point. He reached out and touched it across to Gavrikov, who scored with a seeing-eye wrist shot through traffic.
The Mammoth tied it 1-1 on Schmaltz's power-play goal at 14:29.
Logan Cooley missed wide left on a breakaway, but the Mammoth got the puck back on a forecheck. Shesterkin stopped Clayton Keller's shot from between the circles and Schmaltz scored his fourth of the season on the rebound.
Rangers center Joseph Veleno sustained a lower-body injury midway through the first period and did not return.
