Sedin scored the game-winner at 9:39 of the third period when he took a pass from twin brother Henrik Sedin in the left circle and fired a shot to the far side of Braden Holtby.
The Canucks, who are 2-1 on their seven-game road trip, bounced back from Sunday's loss at Detroit. They haven't dropped back-to-back games since mid-October.
Radim Vrbata added a power-play goal and an assist, and Derek Dorsett also scored for Vancouver. Henrik Sedin had two assists, and Ryan Miller finished with 23 saves.
John Carlson had two goals and an assist, and Brooks Laich also scored for Washington, which is 1-3-1 in its last five games.
Holtby made 28 saves for the Capitals, who have allowed three power-play goals in each of their last two games.
Laich tied the score at 3 during a 4-on-4 sequence early in the third period when he got behind a defender and lifted the puck over Miller's shoulder for his first goal of the season.
Vancouver outshot Washington 12-5 in the first period and took a 1-0 lead with 2:02 left when Jannik Hansen intercepted a clearing pass and got the puck to Bo Horvat in the left circle.
Horvat slid the puck across to Dorsett, who one-timed it past Holtby from the slot for his third of the season.
Carlson tied it at 2:45 of the second period on a shot from the slot off a feed from Nicklas Backstrom.
The Canucks took a 3-1 lead with consecutive power-play goals midway through the period.
With Troy Brouwer off for holding, Vrbata corralled a rebound and sent it around behind the net to Henrik Sedin, who fed it in front. Daniel Sedin beat a sliding Holtby for his fifth of the season at 10:01.
Less than three minutes later, Holtby blocked a shot from Alexander Edler, but Vrbata banged the bouncing puck home for his 11th goal.
Carlson pulled Washington to 3-2 when he netted his second of the period and fourth of the season with a drive from just inside the blue line.
NOTES: The Canucks have won six of their last seven against the Capitals. ... Vancouver last dropped consecutive games on Oct. 18 (Tampa Bay) and Oct. 21 (Dallas). ... The Capitals have only three home games remaining in December. They have a trio of three-game road trips. ... Capitals D Mike Green (upper-body injury) missed his fourth straight game.