Stefan Noesen scored a power-play goal to pull the Devils within 6-5 at 16:45.
John Carlson had three assists, and Dubois, Raddysh and Rasmus Sandin each had a goal and an assist for the Capitals (17-6-1), who won their fourth straight game. Charlie Lindgren made 22 saves.
Washington, which went 3-for-5 on the power play, won 5-4 in overtime against the New York Islanders on Friday. The Devils, who went 2-for-3 on the power play, won 5-4 at the Detroit Red Wings on Friday.
"There were a lot of ebbs and flows in the game," Capitals coach Spencer Carbery said. "I just thought the power play stepped up huge. For us to come in here and get a win, even if it was a little bit odd the way that it played out and both teams off a back-to-back, it's a big game for us."
Jack Hughes had three assists, Noesen scored twice, and Jesper Bratt had a goal and an assist for the Devils (16-9-2). Timo Meier had two assists, and Allen made 26 saves.
"We were just slow to pucks, losing battles," New Jersey defenseman Brett Pesce said. "As a group, myself included, we have to find a way to start better."
Andrew Mangiapane gave the Capitals a 1-0 lead at 6:19 of the first period by jamming in his own rebound. It marked the seventh straight game when New Jersey allowed the first goal.
Nico Hischier tied it 1-1 at 10:54 on a backhand near the right post.
Justin Dowling gave New Jersey a 2-1 lead at 2:38 of the second period, scoring on a redirection from the high slot after a shot from the left point by Pesce.
Washington then took the lead with two power-play goals in a span of 1:07 after Dawson Mercer received a double minor for high-sticking Dylan Strome at 8:49.
"It stings because we take a lot of pride in our penalty kill and tonight it arguably cost us the game," Pesce said.
Connor McMichael tied it 2-2 at 9:49 on a snap shot from the slot on a 4-on-3, and Jakob Chychrun made it 3-2 at 10:56 on a wrist shot from above the right face-off circle on a 5-on-4.
"I just think we're finding the right plays on the power play. I don't think it's overcomplicated," Carlson said. "I think we're just trying to get the puck to guys or to the middle and making things happen."
Sandin pushed it to 4-2 at 19:49 with another power-play goal, on a loose puck in the slot after a scramble behind the net.