Washington tied it 1-1 at 19:27 of the first when Andre Burakovsky beat Lundqvist from the right face-off circle.
Holtby made 35 saves in his first start since March 20, none bigger than during a four-minute Rangers power play after a Michal Kempny high-sticking penalty. Holtby stopped a tipped shot by Mika Zibanejad and a rebound from Vladislav Namestnikov at 15:37 of the second period.
The Capitals killed all four minutes and the game remained tied. New York was 1-for-5 on the power play.
Holtby hadn't allowed fewer than two goals in a game since Nov. 18. He was starting for the fifth time since Feb. 27 between a knee injury and being outplayed by Phillip Grubauer.
Grubauer was scratched because of a lower-body injury after starting four of the past five games.
"I have two goalies," Capitals coach Barry Trotz said. "If that's an indication of where [Holtby's] game is going, I know where Grubauer's is at, then we're in good shape with two goaltenders that have played valuable minutes for us. … We've still got some games left and we'll see where we are."
Eller and Kuznetsov made Holtby's work pay off. Eller stood in front of the net and tipped Nicklas Backstrom's pass over Lundqvist's shoulder with 1:05 remaining in the third period.
On the first shift of overtime, Kuznetsov drove into the offensive zone, circled back as if he was going to carry it back to center ice but then darted toward the net again, faked a slap shot and beat Lundqvist glove side.
"I want to go 1-on-1 so badly over there, but I know we play for quite a while and I've got to save the puck and then did the pull up and no one come [to] me," Kuznetsov said. "[The Rangers] probably tired a little bit too. I don't know what happen with me, I decide to shoot, but it end up working pretty nice."
