Boyle popped in his second goal of the series, and Brassard had his series-leading fifth assist.
"I thought he played a complete game, " Rangers coach John Tortorella said of Brassard.
The Rangers grabbed the lead back with Brassard's power-play goal 1:23 into the second.
Brassard and Boyle both notched their third assists of the game on the goal.
"We are just going to have the same mentality we had coming into this game, " Brassard said.
"We didn't break, " said Rangers forward Derick Brassard, who had a goal and two assists on Monday.
He was glad to know Brassard's leg was saved, too, especially at the hospital where he works.
CNN: In runners' tent and ER, a rush to save limbs -- and lives
Brassard scored once on Monday and had two assists to breathe life back into the Rangers' previously dormant offense.
Lundqvist, captain Ryan Callahan, and Brassard are among those who have one-year remaining before they can become unrestricted free agents.
He freed the puck to center Derick Brassard, who found Arron Ashman for a goal that gave the Rangers a 3-2 lead.
Girardi ripped a shot from above the left circle off a feed from Derick Brassard to give the Rangers the lead again.
Brassard became the first player with four points in his Rangers regular-season debut since Doug Bentley on Jan. 24, 1954, against Boston.
In return, the Rangers received forwards Derek Dorsett and Derick Brassard, defenseman John Moore and a sixth-round pick in the 2014 draft.
Brassard handed Lundqvist the lead at 9:39 of the second period, with a goal that was originally credited to struggling forward Rick Nash.
Segatore, who'd heard Brassard screaming for his daughter in the medical tent, learned from the news that he and his family had survived.
CNN: In runners' tent and ER, a rush to save limbs -- and lives
The infusion of those players, particularly center Derick Brassard, stabilized the roster and helped the Rangers push past the Washington Capitals in the first round.
Derick Brassard was there and made a one-touch pass to Boyle, who had skated in front of the crease with barely a hindrance from Capitals defensemen.
Clowe scored his second of the night and season, again on a power play, with 8:08 left in the second off assists by Brassard and Boyle.
Brassard put the Rangers in front in the second period with the team's first power-play goal in the series, and Asham gave New York its 3-2 lead.
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.
New York used every bit of its second advantage of the first period when Boyle scored exactly two minutes after Joel Ward was sent off for high-sticking Brassard.
Brassard spun in the left circle and backhanded a pass back to Brad Richards at the point for a drive that hit Boyle in front and caromed in.
"We bounced back from 0-2, " Rangers forward Derek Brassard said.
Henrik Lundqvist made 27 saves and Derick Brassard provided the only offense in New York's 1-0 victory that set up a decisive Game 7 in Washington on Monday night.
New York continued its onslaught in the second period, pushing the lead to 4-0 just 1:53 in when Brassard scored his first with the Rangers on a power play.
Nicklas Backstrom gave the Capitals a 1-0 lead 4:06 in, but Brassard put the Rangers in front in the second period with the team's first power-play goal in the series.
Nash recorded only one goal and four assists in this year's playoffs, and even though Brassard led the club with 12 postseason points, the big scoring threat of Gaborik was missed.
Fleury's best stop came with 5 minutes to go when he lost his balance and fell on his backside as Brassard threw a shot from behind the net at Fleury's legs.
Brassard, a pickup from Columbus in the deal that sent Marian Gaborik to the Blue Jackets at the trade deadline, also had five assists in the first five games of the series.
"The fact that we came back here in our building helps a lot, " said the 25-year-old Brassard, who had a goal and three assists in his Rangers debut April 3 against Pittsburgh.
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