Cutter Gauthier also scored for the Ducks (17-10-1), who have alternated wins and losses over the past nine games (5-4-0), including a 7-0 loss to the Utah Mammoth on Wednesday. Ville Husso made 18 saves and stopped two of three attempts in the shootout.
Troy Terry and Mason McTavish scored in the shootout for Anaheim.
Tom Wilson, Ethen Frank and Aliaksei Protas each scored for the Capitals (17-9-3), who finished their four-game road trip 3-0-1. Logan Thompson made 35 saves.
Wilson gave the Capitals a 1-0 lead at 9:29 of the first period. Trevor van Riemsdyk's shot from the right boards hit Justin Sourdif in the right face-off circle. Sourdif then tapped the loose puck to Wilson, who beat Husso blocker side with a snap shot from the high slot.
Washington rookie forward Ryan Leonard left the game after taking a hit from Ducks defenseman Jacob Trouba at 16:04 of the first period. The play was reviewed for a major, but the only penalty assessed was to Capitals defenseman Jakob Chychrun for roughing.
Gauthier scored from the slot off a feed from Sennecke from below the goal line 12 seconds after the penalty expired to tie it 1-1 at 18:16.
Frank put the Capitals back up 2-1 at 6:06 of the second period. A clearing attempt by Ducks defenseman Pavel Mintyukov went between his legs directly to Frank in front, who lifted a shot past Husso's glove.
The Ducks scored just 13 seconds later when Johnston tipped in a long-distance shot from Radko Gudas to even it 2-2 at 6:19.
Protas gave Washington a 3-2 lead at 7:38, burying a rebound that came off Husso's left pad off a point shot from Matt Roy.
Sennecke tied the game 3-3 at 17:59. Thompson was unable to control a pass back to him from Martin Fehervary and Johnston made a short pass to Sennecke. Sennecke's shot from in tight went off the stick blade of Washington defenseman Declan Chisholm and over Thompson's glove.
