After the Capitals won a draw deep in the Boston zone, Marcus Johansson got the puck behind the net and fed Backstrom in the right circle, who fired a one-timer past Bruins netminder Tim Thomas for the game-winner.
Troy Brouwer scored in the second period for the Caps, who bounced back from a 1-0 defeat in Thursday's Game 1 of the Eastern Conference series.
Braden Holtby turned aside 43-of-44 shots while logging his second straight strong game for Washington.
Boston's Beniot Pouliot dented Holtby with a goal 12:13 into the final period for the equalizer, forcing overtime.
Thomas finished with 37 saves for the Bruins.