New York generated offense with its struggling power play that had failed to score on seven chances in Washington.
The Rangers used every bit of their second advantage of the first period, when Boyle scored exactly two minutes after Joel Ward was sent off for high-sticking Brassard.
Boyle, who returned to the lineup for Game 2 on Saturday following a knee injury, carried the puck from the right-wing wall, cut into the circle, shifted the puck to both sides of his stick, and flipped a shot over a crouched Green and past Holtby at 12:50 for what was technically an even-strength goal.
That got the Rangers even and took some of the nervous tension out of the crowd that had already let out loud groans after Backstrom gave the Capitals a 1-0 lead when he deflected in John Carlson's shot in the slot.
"We accomplished what we set out to do," Boyle said. "We could only control Game 3. It was by no means easy. They came hard, but it was a good win for us. We can enjoy it for a little bit, but we have to make it stand up."
The Rangers grabbed the lead back with Brassard's power-play goal 1:23 into the second. 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.
It took only eight seconds of power-play time for the Rangers to cash in. Mats Zuccarello sent a pass from near the right point into the slot to Brassard, who beat Holtby with a quick drive and then flashed an emphatic fist pump to celebrate his first NHL playoff goal.
Lundqvist was strong throughout the second period and did his part to keep the Rangers in front.
After heavy pressure by the Capitals in the New York end, Lundqvist made a pad save on a point-blank drive from the slot. That didn't end the Washington attack, as Green let go a shot that Lundqvist snared with his glove to get a stoppage of play.
It also elicited a healthy chant of "Hen-rik, Hen-rik" from the crowd that sensed every save was crucial because of the Rangers' slumping offense.
"There was a lot of determination going into this game," Lundqvist said. "We knew we had to win this one. They're a good team, and you have to respect that, but we scored some big goals."
Copyright 2013 The Associated Press. All rights reserved. This material may not be published, broadcast, rewritten or redistributed.