Each team tallied twice in the shootout before Anisimov ended the duel by beating Braden Holtby, who was making his seventh career start.
The last time the teams met, on Dec. 12 in New York, the Rangers rolled to a 7-0 win. In this one, the Capitals cranked up the defense and took a 1-0 lead in the second period on a goal by Matt Hendricks.
Marian Gaborik tied it for the injury-riddled Rangers with 6:41 left in regulation. Gaborik was in front of the net when Brian Boyle took a shot, and the rebound bounced off Gaborik and past Holtby.
"I felt it hit me and then I turned away and saw the guys celebrating," Gaborik said. "I didn't have any idea. I guess it was one of the lucky ones."
That's how the Capitals felt, too.
"It was a lucky bounce, a fluke goal on their part," Hendricks said.
And, it ended Holtby's bid for his first NHL shutout.
"The only goal that beat him was a goal that even the Rangers didn't know they scored," Capitals coach Bruce Boudreau said.
New York improved to 8-3 in games that extend beyond regulation. Washington lost its seventh consecutive overtime game and ninth in the last 10 despite outshooting the Rangers 6-1 in the extra five minutes.
"We haven't lost the overtime games because of lack of offense; it's because of poor defense," Boudreau said. "We talked about how we have to play in the overtime and we talked about it again on the bench. I thought we did a real good job of it."
Biron made several sensational saves in overtime, including one on a blast by Alex Ovechkin, who was held without a point.
"Tonight I thought he had two really good chances to score," Boudreau said. "Sometimes you get them. Some years they're all going in, and some years they're not going in. He's fighting for every goal he can get this year."
Holtby finished with 28 saves. In three games since his recall from Hershey of the American Hockey League last week, the 21-year-old has turned aside 77 of 80 shots.
"NHL shootouts are a lot harder, obviously," Holtby said. "It's not something you get to practice because when they clean the ice it's a lot more slippery and you usually practice shootouts at the end of practice when it's all chewed up."
The focus was on defense during a scoreless first period in which the Rangers outshot Washington 7-5. Each team took only one shot over the first 11 minutes, and that included a Capitals power play in which Biron was not required to make a save.
At 1:27 of the second period, Hendricks scored from the left side of the net off a brilliant diagonal pass by Marcus Johansson. It was the seventh goal of the season for Hendricks, his second in two games.
New York scratched defenseman Dan Girardi, who missed his first game of the season with a rib cage injury. The Rangers are trying to keep pace in the Eastern Conference while coping with injuries to several key players, including leading scorer Brandon Dubinsky, who has a stress fracture in his left leg.
"We grind. We have confidence we can wear teams down," Boyle said.
"It's going to be like this for the next 2 1/2 months," coach John Tortorella said. "We're happy we're grabbing some points here."
Material from wire services and team media was used in this report.
1 - 0 WSH