Shane Pinto, Nick Cousins and Thomas Chabot also scored for the Senators (4-4-1), who are 3-1-1 in their past five games. Linus Ullmark made 12 saves.
Trevor van Riemsdyk scored for the Capitals (6-3-0), who won 5-1 at the Columbus Blue Jackets on Friday. Charlie Lindgren made 27 saves.
Alex Ovechkin became the eighth player in NHL history to play 1,500 games with one franchise. He remains one goal away from becoming the first player to reach 900 in the NHL.
Cozens gave the Senators a 1-0 lead at 1:58 of the first period when he redirected a pass from Artem Zub past Lindgren from the slot.
Pinto made it 2-0 at 3:27 of the second period when he stole the puck, came in on a partial breakaway and poked the puck in. The net came loose after he collided with Lindgren, and after a review the goal was upheld as the puck crossed the goal line before the net was displaced.
The Capitals challenged for goaltender interference, but it was determined that Washington defenseman Jakob Chychrun pushed Pinto into Lindgren.
On the ensuing power play, Cozens increased the lead to 3-0 when he took a back-door pass from Batherson, deked around Lindgren at the net front and lifted in a backhander at 4:27.
Cousins extended it to 4-0 at 6:22 when he was sent in on a breakaway by Stutzle and scored.
Batherson made it 5-0 when he knocked in a rebound during a power play 55 seconds into the third period.
Van Riemsdyk scored during a goal-mouth scramble at 5:04 to cut it to 5-1.
Chabot increased the lead to 6-1 with a power-play goal through traffic at 9:47. Batherson scored at 14:22 for the 7-1 final.
Capitals forward Dylan Strome left the game in the first period with a lower-body injury.
