Panarin made it 2-0 at 19:49, scoring a power-play goal off a short-side shot from the left circle.
Shesterkin made 15 saves in the first period.
"They controlled the first period a little bit, but we took advantage of our chances," Rangers coach Gerard Gallant said. "Igor made some good saves, probably frustrated them a little bit, and we buried our chances. That was the difference in the game."
Tarasenko scored the first of four New York goals in the second period at 3:54, making it 3-0 on a one-timer off a back-and-forth passing sequence with Zibanejad during a 3-on-1 rush with Panarin on the far side.
Kreider then made it 4-0 at 6:43, darting to the net and redirecting Kane's backhanded pass through the slot past Jarry.
"You know 'Kreids' drives the net," Kane said of Kreider. "I tried to chip it over at first and it got blocked, but I knew he was going backdoor, so you just try to get it to him on his forehand. He's not going to miss too many of those."