Tanguay turned to the right as he bore in on Auld and skated backward before scoring the winner after Andrei Markov had the only goal on Montreal's first three attempts.
"It's a move that either the goalie plays the shot there or he opens his legs to cover both posts," Tanguay said.
Auld acknowledged that skaters are using an ever-growing range of approaches on shootout attempts.
"Guys are trying anything and that was a pretty good play," Auld said. "I didn't know whether he was going to pull it back to his forehand or not. I was kind of thinking that if he was going to shoot, he'd probably go high -- on the backhand, usually guys try to get it up. It was a good, patient play on his part."
The Senators' losing streak reached six games (0-4-2).
"We're not happy with it, but I think the way we played is encouraging," Ruutu said. "If we keep working like we did tonight, playing the way we did, we're going to get rewarded for it. You can't bury your head and feel sorry for yourself. It's actually the opposite, we should feel a lot better than we did a couple of games ago."
Markov drew Montreal even at 2 with 4:24 remaining in the third. The Russian defenseman drove a slap shot past Auld from the right faceoff circle for his second goal of the season.
"It was a pretty tight game, I thought," Tanguay said. "It was a fast-paced game and both sides were desperate to get a win. It's a huge relief for us. Andrei has played some great hockey for us. The puck hasn't gone in as much as he'd like but he made an outstanding shot there and fortunately for us, that puck found its way into the net."
Saku Koivu pulled the Canadiens even at 1 with his seventh goal on a power play 13:33 into the first.
"It was a huge win for us, obviously," Canadiens coach Guy Carbonneau said. "Both teams started the game knowing that it was going to be a tough game, that both teams were kind of struggling lately and that everything was going to be left on the ice. It was a really good game."
Auld stopped 19 shots in his fifth straight start for Ottawa.
Montreal, which lost 2-1 in Carolina on Tuesday, won for just the third time in its last eight games (3-4-1) since getting off to an 8-1-1 start.
"I thought this was a step in the right direction for our team," Price said. "I thought the last three games we started to turn it around and start to play well, even though we lost the last game. We're starting to bond more as a group, I think."
Fisher strained a ligament in his right knee in Monday's 2-1 shootout loss in New York and is expected to be out for a week. Neil will have arthroscopic surgery Friday to repair a meniscus tear in his right knee. He's expected to miss at least a month.
Heatley opened the scoring when he flipped a shot past Price from the slot after Alfredsson redirected Alexandre Picard's point shot.
Koivu jammed a shot between Auld's pads from the right edge of the crease at 13:33 to tie it at 1.
Antoine Vermette appeared to have scored the go-ahead goal for Ottawa 17:55 into the second during a delayed penalty call.
Referee Don VanMassenhoven immediately waved it off, despite Senators alternate captain Chris Phillips' protests, indicating that he had blown his whistle when Montreal defenseman Francis Bouillon played the puck just before Vermette beat Price with his shot from the slot.
"I heard it right before he released the puck," Price said.
- Canadiens C Maxim Lapierre and RW Mathieu Dandenault were healthy scratches.
- Senators C Cody Bass made his season debut on a line with Foligno and Ilya Zubov.
- Veteran D Luke Richardson did not play for Ottawa.