Huet, who stopped Alfredsson on a penalty shot in the second, came within 5:10 of recording his second shutout in as many starts before the Senators captain tied it at 1 with his 12th goal of the season.
Alfredsson took Dany Heatley's pass from the right side and beat Huet with a shot from the slot.
"You've just got to play smart and you can't force things," Alfredsson said. "If you start thinking, 'Oh my God, what's going to happen if we don't win?', that's when you get in trouble. You go on instinct, you go from what the coaches have told you to do out there and do the little things right. We knew we were going to create chances."
Neil gave Ottawa the lead when he beat Huet with a shot from the right faceoff circle with 3:30 left for his second goal.
"You could feel the momentum changing in the third period and we just had our foot on the gas pedal and kept going," Neil said.
Alfredsson added an empty-net goal for his 13th of the season -- his fourth in three games -- as the Senators improved to 14-2-0. Ottawa lost 4-1 to Washington on Thursday to end its team-record eight-game winning streak.
"Overall, I thought it was a heck of a game," Alfredsson said. "It was very up-tempo and both goalies played really well. Both teams gave it all they had out there and we were really lucky to come out on top."
Martin Gerber stopped 28 shots for his league-leading 11th win.
"Everyone's talking about the way Huet played -- he played unbelievable but our goalie played unbelievable, too," Neil said. "He stepped up and played another solid game for us and he's done it all year."
Huet, who was rested for Thursday's 2-1 win in Boston, made 34 saves. He stopped 29 shots for his first shutout in nearly a year in Monday's 2-0 win over Buffalo.
"It's a hard one, not getting any points," Huet said. "Everybody played very well but you know, we can definitely play with them. The season is long and I'm sure it's going to be a lot of good hockey against that team for the rest of the season."
Andrei Markov scored his fifth goal of the season on a power play 11:11 in and the Canadiens nearly made it stand as the game's only goal.
Montreal, which has won three straight, went 1-for-4 with the man advantage. The Canadiens have scored 23 goals in 78 power-play opportunities for a league-leading 29.5 percent success rate.
Markov drove a slap shot from the point over Gerber's right shoulder and into the top left corner with 8:49 remaining in the first, 38 seconds after Ottawa's Chris Neil was sent off with a delay of game penalty for clearing the puck over the glass.
"I don't think [Koivu] wanted to," Huet said. "We're not going to forget that, somebody going after our captain and top player."
Huet made a remarkable glove save on Randy Robitaille's shot from the right side in the second and denied Alfredsson's backhand attempt on his penalty shot at 10:53.
Montreal missed a couple of good opportunities to add to its lead as Chris Higgins hit the left post in the dying seconds of the second period, and Koivu put a backhand off the post in the third on a play that necessitated a video review.
Canadiens C Steve Begin was out of the lineup because of a charley horse. D Josh Gorges also didn't dress for Montreal. ... Senators C Jason Spezza missed his sixth straight game because of a groin injury. ... RW Brian McGrattan was Ottawa's only healthy scratch. ... A sellout crowd of 20,065 was on hand for the game.