Bitetto gave New York a 2-0 lead at 9:15 of the second period with what he called a "blackout moment" goal.
The defenseman carried the puck behind the net, regathered it in the corner after getting hit by Hagelin, and went up the left-wing boards. He avoided a hit from Washington forward Nic Dowd, cut to the middle, eluded a stick check from Capitals defenseman John Carlson and beat Vanecek with a backhand from between the hash marks.
It was his first goal in 113 games, since Nov. 18, 2017 with the Nashville Predators.
"It was magical," Bitetto said. "I usually don't score those type of goals. You put that [Rangers] jersey on, weird things happen."
Bitetto grew up a Rangers fan in Island Park, New York, 30 miles from Madison Square Garden. It was his third NHL goal in his 185th game, including the past two with the Rangers.
"It's funny, my last three goals were disallowed," Bitetto said. "I scored that tonight and my mindset was just like, 'Please allow it, don't challenge this one. I need it.' Thank God [they] didn't."
Washington cut it to 2-1 on Hagelin's goal at 18:57 of the second.
New York forward Brett Howden was called for slashing with 10 seconds left in the second period, but the Rangers killed the penalty to start the third. They were 4-for-4 on the penalty kill and are 14-for-14 in the past three games.