The Rangers had taken a 1-0 lead in the first off a gaffe by Holtby.
Braden Holtby stopped 20 shots through overtime, and didn't allow a goal in the shootout.
After Holtby denied Brad Richards, Backstrom beat Lundqvist to decide the contest in the fourth round.
New York generated a season-high 20 shots overall, but was denied each time by Holtby.
Washington squandered a 5-on-3 early in the period and Holtby stopped Marian Gaborik's breakaway bid midway through.
He made more drives to the net in an effort to create more traffic in front of Holtby.
Holtby and Lundqvist were both on top of their games Saturday and pitched regulation shutouts at each other.
The Rangers pressed hard for the winner at the start of the extra period with Callahan testing Holtby.
Washington couldn't tie it again, even after pulling Holtby for an extra skater.
Holtby is making a name for himself, too, after going 23-12-1 with a 2.58 goals-against average in 36 regular-season games.
With Holtby well out of position, Hagelin's shot at a net guarded only by defenseman John Erskine was saved by Erskine.
From behind his net, Holtby swept the puck up the left boards.
"It's a credit to our guys for sticking with it, " Holtby said.
That winner was scored by Derek Stepan, who deflected a pass from Nash in behind goalie Braden Holtby with 6:25 left in regulation.
Until then, New York had been stymied by the Capitals and goalie Braden Holtby, who had yielded just one goal in two games.
"We know they're going to make a push, " Holtby said Sunday after the Capitals held an optional practice before heading to New York.
No, Washington's Braden Holtby doesn't have the awards, the accolades or the attention that New York counterpart Henrik Lundqvist has grown accustomed to.
After Holtby got his right pad out to stop Buffalo's Tyler Ennis, Ovechkin sealed the victory by snapping a shot through Enroth's legs.
The Rangers cashed in on a 5-on-3 advantage for the equalizer at 18:03 as Stepan scored from a sharp angle to Holtby's right.
Until then, New York had been shut down by Holtby, who had suggested he wasn't tested much in Washington's 1-0 overtime victory on Saturday.
Rookie Chris Kreider left a drop pass for Arron Asham, whose slap shot from the right circle glanced off Holtby's glove and into the net.
He then covered the puck in front with 48.4 seconds left, keeping Marcus Johansson at bay after the Capitals pulled Holtby for an extra attacker.
Nash was more of a presence in front, making more drives to the net in an effort to create more traffic in front of Holtby.
Until then, New York had been shut down by goalie Braden Holtby, who had suggested he wasn't tested much in Washington's 1-0 overtime victory on Saturday.
While Callahan did have 16 goals this season, the other four Rangers who put pucks past Braden Holtby on Monday combined for a total of only 14.
What made this one even more unique was the fact New York got the advantage when Holtby tripped Nash as he skated out from behind the net.
Holtby, who suggested he wasn't tested much in Washington's 1-0 overtime victory on Saturday, had plenty to deal with when the series shifted to Madison Square Garden.
After the Rangers' third goal, Oates pulled Holtby in favor of Michal Neuvirth, who hadn't played since Feb. 7 and had recently been sidelined with an illness.
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