Joel Ward snatched the game-winner with a backhanded rebound in the extra session, and Caps rookie goaltender Braden Holtby made 31 stops to get the better of his more accomplished Bruins counterpart Tim Thomas.
"He's unbelievable. He's carried the load for us," Ward said of the 22-year-old Holtby. "He's calm and cool every day. Hopefully he takes us over through the next round."
Each of the seven games in the series were decided by a single goal, a first for the NHL play-offs, with four of them going to overtime.
Matt Hendricks scored for the Capitals in the first period before Boston's Tyler Seguin made it 1-1 in the second and the teams battled through a scoreless third period.
Washington dictated most of the action in the third but had to kill off a penalty when Jason Chimera was sent to the box with 2:26 left in regulation.
After keeping the Bruins at bay and forcing overtime, Mike Knuble picked up a turnover and raced to the front of Boston's net where Ward followed in and finished off Knuble's shot to extend Washington's season and put an end to Boston's.
"It's hard to swallow, tough to understand right now," said Bruins center Patrice Bergeron. "We obviously weren't ready for being done right now."
The unexpected emergence of Holtby has the Capitals aiming for a first conference final since 1998 and comes after Washington failed to win their division after four consecutive Southeast titles.
"Tonight as a group we felt we had nothing to lose and the pressure was kind of on them," Holtby said.
"It is my first ever (play-off series win) in my career and it feels good to be in the celebration instead of watching it, that's for sure."