Smith set up Adam Fox for a tip-in attempt as the Rangers defenseman came across the crease, but Tarasov held his ground for the save and cover at 14:25 of the second.
Earlier in the second period, K'Andre Miller had an open look from the slot off a rebound of Zibanejad's shot, but Tarasov moved across in time to make the save at 4:19.
In the first period, Shesterkin went post to post to stop Cole Sillinger on a redirect with his right pad at 4:40.
Chris Kreider deflected Fox's wrist shot off the right post during a Rangers power play at 11:14.
"They've got skilled guys that are going to get chances, but after the first 10 [minutes] we settled in, had some looks ourselves," Fox said. "Sometimes a boring game instead of a track meet is good. We did a good job of that today."
Credit Tarasov's saves on Zibanejad's breakaways in overtime for the game getting to a shootout.
"He made some great saves on the breakaways that we gave up, that we've got to obviously look after a little bit," Evason said. "He played a strong game, and I think everybody was committed in front of him as well."
But when it got to the shootout, Shesterkin remained on point. Trocheck, though, said Tarasov gave him blocker side so he took it and scored in the second round of the shootout.
"I was just going down to see what I can do," Trocheck said.
Shesterkin stopped Fantilli after Trocheck scored and his save on Marchenko in the bottom of the third round sealed the win for New York.
"I just saw he did a step right and I know righty players like to do high glove," Shesterkin said, "but the puck a little bit jumped in front of his stick, and I took the chance."
NOTES: Shesterkin's shutout was the 17th of his career. He passed John Vanbiesbrouck (16) to move into ninth place all-time among Rangers goalies.