The Rangers played the kind of defense that they're going to need come playoff time.
Derek Brassard also scored for the Rangers, who had two short-handed goals in the game to move two points up on the Philadelphia Flyers for second place in the Metropolitan Division.
"On some nights you need some bounces and we got a couple tonight," Rangers coach Alain Vigneault said. "Our penalty killing gave us two big goals tonight, which was good to see.
"We hadn't played as well as we would have liked defensively in (a 4-3 loss to) Calgary on Friday. We were a lot better tonight."
Center Derek Stepan, who had three assists in the game, agreed.
"We tightened it up and moving forward we have to play that way because the points are so big," Stepan said. "We have played a good defensive game over the last few weeks and that is what is going to lead to our offense and help us win."
The Rangers tied their franchise record for road wins with 24 this season.
The Oilers have lost four of their last five games.
Edmonton's Ryan Nugent-Hopkins said a team can't surrender two short-handed goals in a game and expect to have success.
"We've given up the most short-handed goals this season and it's something that we can't afford to do," he said.
It was the ninth time this season that the Oilers have been shut out.
Oilers coach Dallas Eakins said his team's lack of offense is something the organization hopes to address in the offseason.
"There is a misconception about this team," he said. "This is not a high-powered offensive team. It's not. The history over the last four or five years knows that. We have to find a way to manufacture some offense and it's something that we're going to have to look into this summer as we add and subtract."
New York took a 1-0 lead with 3:30 to play in the opening frame when Zuccarello tipped in Anton Stralman's shot on a play that goalie Ben Scrivens argued involved a high stick. It was reviewed and Zuccarello got his 16th goal of the season.
The Rangers had eight first-period shots, while the Oilers mustered three on Talbot.
New York took a 2-0 lead 4:38 into the second period on the power play, when Brassard's knuckler from the sideboards seemed to catch Scrivens off guard.
The Rangers took a three-goal advantage 9:07 into the period when Nash threw a backhand from behind the goal-line and it hit off Scrivens and went in for Nash's 24th goal of the season.
New York made it 4-0 with 6:34 left in the second period on a short-handed two-on-one break with Nash converting a pass from Derek Stepan.
The Rangers upped their lead to 5-0 on another short-handed goal four minutes into the third period when defender Philip Larsen overcommitted to covering Stepan. He was able to earn his third assist with a feed to Zuccarello in front of the net for his second goal. It was the 13th short-handed goal the Oilers have allowed this season.
Both teams return to the ice on Tuesday. The Rangers play the third game of a four-game trip in Vancouver and the Oilers take their final three-game road trip of the season, starting in San Jose.