Washington put together a three-goal second to win for the fifth time in seven games.
Marcus Johansson and John Carlson both scored during the opening nine minutes of the middle stanza to give Washington a 2-1 lead. Riley Nash then tied it for the Hurricanes on the power play before Brouwer responded with his seventh of the season.
Philipp Grubauer, in his sixth appearance of the season, protected the one-goal lead until Alex Ovechkin scored into an empty net with 25 seconds to play. It was the league-leading 30th goal of the season for Ovechkin.
Grubauer, whose record improved to 4-0-0, made 40 saves in the well-earned victory. Cam Ward surrendered three goals on 28 shots.