Braden Holtby stopped 43 shots for Washington in his second career postseason game.
Tim Thomas made 37 saves for the defending Stanley Cup champions.
Troy Brouwer opened the scoring for the Capitals in the second period, and Benoit Pouliot tied it with 7:47 left in regulation.
On the winner, Marcus Johansson got the puck from behind the net and passed it to Backstrom at the left faceoff circle. He wristed it past Thomas on his stick side.
Boston won the opener 1-0 in overtime Thursday night.
Game 3 is Monday night in Washington.
SENATORS 3, RANGERS 2 (OT)
In New York, Chris Neil scored 1:17 into overtime, and the Ottawa Senators rallied for a wild victory in Game 2 to even the first-round Eastern Conference series
Neil got to a loose puck in front off a rebound and backhanded the puck past goalie Henrik Lundqvist.
The top-seeded Rangers seemed poised to grab a 2-0 series lead when Brian Boyle scored in the third period, but Nick Foligno tied it with 4:37 left in regulation.
Game 3 of the best-of-seven series is Monday night in Ottawa, followed by Game 4 on Wednesday night. The series will return to New York for Game 5 next Saturday.
SHARKS 3, BLUES 0
In St. Louis, Brian Elliott and Jaroslav Halak combined for a shutout, the San Jose Sharks put themselves in an early hole when Marc-Edouard Vlasic tapped the puck into his own net and the St. Louis Blues evened the first-round series at a game apiece.
Vladimir Sobotka was credited with a goal on the first shot of the game after Vlasic’s gaffe.
David Backes and David Perron also scored, and T.J. Oshie had two assists in the Blues’ first playoff victory since 2004.
Halak and Elliott combined for 15 shutouts in the regular season, tying a modern NHL record.
They won the Jennings Trophy for the fewest goals allowed.