Brouwer gave the Capitals a 3-2 series lead with his second goal of the playoffs, beating goalie Tim Thomas over the glove with a wrist shot from the right circle. It came with 37 seconds left on a slashing penalty against Benoit Pouliot.
The Bruins, who trailed 2-0 and 3-2, tied it at 8:47 of the third period on Johnny Boychuk's goal.
The Capitals can wrap up the best-of-seven series at home Sunday.
The Bruins have been in this predicament before. They trailed the Vancouver Canucks 3-2 last year in the Stanley Cup finals, then won the last two games on the road to capture their first title since 1972.
Washington took a 2-0 lead in the second period on goals by Alexander Semin at 11:16 and Jay Beagle at 14:27. That seemed to be a sizable advantage in a series that featured tight defense with each team scoring only seven goals in the first four games.
But the Bruins rallied with two goals in 28 seconds late in the period to tie at 2. Dennis Seidenberg scored with 2:39 left on a one-timer from the right circle that got past goalie Braden Holtby. And Brad Marchand tied with 2:11 to go when he poked the rebound through Holtby's legs after a shot from the right point by Boychuk.
Washington regained the lead 3:21 into the third period when Mike Knuble pounced on a rebound. Joel Ward shot from 50 feet on the right side and Thomas saved it. But the goalie steered the puck to the left side and couldn't slide over in time to stop Knuble's shot.
Boston rallied again when Boychuk scored on a 50-footer from the left to the far side past Holtby's glove.
Holtby finished with 34 saves, and Thomas stopped 28 shots.
Notes: Seidenberg's goal was the Bruins first in 84 minutes, 9 seconds since Rich Peverley scored in the first period of the Capitals' 2-1 victory Thursday night. ... Boston coach Claude Julien shook up his lines to try for more offense and it finally paid off late in the second period. ... Beagle's goal was his first career playoff point. ... Washington's 2-0 lead was the first two-goal advantage for either team in the series.