Zibanejad gave New York a 1-0 lead with a short-handed goal at 3:25 of the second. He took a stretch pass from Jacob Trouba and faked a shot from the right circle before cutting across to roof a backhand over Vasilevskiy.
Trouba, who was named Rangers captain Aug. 9, finished with one assist, six shots on goal, three hits and two blocked shots in 25:49 of ice time.
"Everyone loves him; he's a great player, great teammate," Trouba said of Zibanejad. "He had a great playoffs (led Rangers with 24 points in 20 games) and obviously coming out and starting out like that, he's a special player. … That was exciting. We were all waiting to get back here and start the season with all the talk and everything that happened over the summer."
Goodrow extended the lead to 3-1 at 11:00 of the third, tipping in a shot from Ryan Lindgren in front of the net.
"This is what I expected from our group tonight, it really is," New York coach Gerard Gallant said. " … Our team looked like a different team than we did last year early in the year, so hopefully we can continue that."