Ryan Strome scored his 100th NHL goal and had an assist for New York (6-7-3), which has won two in a row after an 0-3-1 skid. Artemi Panarin had two assists.
Zibanejad ended his drought with a shorthanded empty-net goal at 17:48 of the third period to give New York a 4-1 lead. He led the Rangers with 41 goals last season.
"If you look at his reaction, you sensed relief, and you look at reaction from some of the guys on the ice, there was relief," Rangers coach David Quinn said. "Listen, he's getting out of it. It's not going to happen overnight. He's not going to go from where he is to where he was last year. We just need to continue to see progress and thought that was another step in the right direction today."
Dmitry Orlov scored, and Vitek Vanecek made 20 saves in his 13th consecutive start for Washington (8-5-3).
"It was tight out there today. The shots were low, the attempts were low, the scoring chances were low," Capitals coach Peter Laviolette said. "We tried to change the lines up going into the third period, we swapped a couple of centerman just to see if that might loosen it up, try to play a faster game, try not to slow it down quite as much. We couldn't seem to get ahead of it."
Quinn shuffled the lines, including putting Zibanejad on a line between Alexis Lafreniere, the No. 1 pick in the 2020 NHL Draft, and Panarin.
"I probably might've put these guys (Zibanejad and Panarin) together earlier," Quinn said. "I liked our balance on our lines today, and listen, if you don't change things, things stay the same. Maybe I should've done it sooner. I certainly had been contemplating it, and I did like the balance and the look of our lines today."
Chris Kreider gave the Rangers a 1-0 lead at 14:57 of the first period when he knocked in the rebound of Panarin's shot on a power play.
Lafreniere made it 2-0 when he took a pass alone in the slot from Panarin and lifted a backhand over Vanecek's pads at 17:44 of the second period for his second NHL goal.