Braden Holtby, Washington's 22-year-old rookie netminder, was brilliant in goal, repeatedly stopping Boston drives to the net with acrobatic saves.
The series, with each game having been decided by just one goal, heads back to Boston for Game Five on Saturday.
Washington scored the game-winner with just 1:17 left in the second period when Alexander Semin blasted a shot from the left wing circle past Boston's veteran netminder Tim Thomas.
Semin's goal gave Washington a 2-1 lead and Holtby thereafter enabled the Capitals to preserve the lead.
Washington set the tone just 1:22 into the game when Marcus Johansson flicked a wrist shot into the net, beating Thomas high to the glove side for his first goal of the series.
The score ignited the crowd of nearly 19,000 and vanquished any fears the Capitals would suffer any fallout from the loss of Backstrom, who was suspended for cross-checking Rich Peverley in the face at the end of Boston's 4-3 victory on Monday.
Peverley tied the score at 1-1 with less than seven minutes left in the opening period when he fired the puck from the right wing circle through the legs of Holtby.