Backstrom got his fourth assist when Kuznetsov scored at 1:10 of the third period to make it 5-2.
"It's tough to watch hockey from upstairs, and it's so good to get back on the ice and even better when you win," Kuznetsov said.
Jakub Jerabek made it 6-2 at 4:52 with his first goal in seven games since being traded to Washington by Montreal on Feb. 21.
Hudon made it 6-3 with his second goal of the night at 9:28, and Gallagher scored on the power play with 1:57 remaining to make it 6-4.
Alex Galchenyuk gave Montreal a 1-0 lead at 8:35 of the first period.
Gallagher, who played his 400th NHL game, appeared to score at 9:38, but Washington coach Barry Trotz challenged for offside, and the goal was overturned after video review showed Jonathan Drouin entered the zone ahead of the puck at 9:25.
Philipp Grubauer, who has started seven of nine games for the Capitals, made 17 saves.