Nigel Dawes also scored in the shootout for the Rangers, but Alex Ovechkin forced the extra round with a goal.
Despite going 0-for-6 on the power play - including five chances in the scoreless third period - the Rangers shook off two blown leads and rallied to win.
Lauri Korpikoski, Markus Naslund and Paul Mara also had goals for the Rangers, who hadn't scored more than three in their previous six games and had been outscored 22-5 during their skid (0-4-1). Henrik Lundqvist made 26 saves.
Tomas Fleischmann and Eric Fehr scored first-period goals for the Capitals, who also led twice. Theodore stopped 28 shots.
The game broke wide open in the second period when the teams combined for five goals yet finished the frame in a 4-4 tie.
New York struck first to get even at 2 and did it with its first shot of the period 8:49 into it. Marc Staal sent in a drive from the left point that struck Korpikoski. The Rangers rookie found the puck and knocked in his fourth goal.
Green put the Capitals back in front with a hard, rising wrister from just inside the top edge of the right circle at 10:32. That tied the record set by Boston's Mike O'Connell, who scored in seven straight games during 1983-84 season.
Ovechkin, the Capitals' leading scorer this season, retrieved the puck for Green and flipped it into the Washington bench for safekeeping.
Naslund needed just 1:07 to bring the Rangers into the third tie of the game, and he did it by following the puck into the net with a skates-first slide. Nigel Dawes raced the puck up the right wing and fed a pass into the crease that Naslund steered into Theodore with his stick. He fell at the same time and pushed himself and the puck into the net, scoring a goal that stood up to video review.
Mara staked the Rangers to their second lead of the night with four minutes left in the period, beating Theodore with a hard slap shot from above the left circle to make it 4-3.
Green struck again with his career-best 21st goal. From the right-wing boards, Green beat Lundqvist with another high drive to get the Capitals even at 4 with 57.6 seconds left in the period.
He has two goals in two consecutive games and has posted six straight multipoint efforts.
The Rangers came out with their fists flying and they looked to send a message after losing a season-high five straight games. Heavyweight Colton Orr took on Capitals counterpart Donald Brashear, considered the toughest fighter in the NHL, right off a faceoff just 2:43 into the game.
Just 3 seconds later, after the ensuing draw, New York's Aaron Voros dropped the gloves with Matt Bradley in a bout that had players on both benches smacking their sticks against the boards and ice.
Callahan gave the Rangers a 1-0 lead 7:33 in, when he took a perfect cross-ice feed from Korpikoski near the right post and easily steered the puck in for his 13th goal.
Fleischmann tied it 1-1 at 10:51 and then earned the primary assist on Fehr's goal with 1:52 remaining in the first period that gave the Capitals their first lead of the game.
Notes: Green, who leads NHL defensemen with 21 goals and 49 points, has nine goals and seven assists during his streak. ... Punch Broadbent holds the overall NHL record for consecutive games with a goal, scoring in 16 straight with Ottawa during the 1921-22 season. He had 27 goals during his run. ... Rangers are in a 1-for-23 power-play slump.
