Canadiens goalie Cristobal Huet, who stopped Alfredsson on a penalty shot in the second, came within 5:10 of his recording his second shutout in as many starts before the Senators captain tied it at 1-1 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.
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.
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.
Martin Gerber stopped 28 shots for his league-leading 11th win.
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.
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 per cent 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.
Canadiens captain Saku Koivu exchanged blows with Ottawa's Patrick Eaves at 14:15. Both were given double roughing minors.
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.
Notes: 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 ... The Senators don't see action again until Thursday, when the Montreal Canadiens visit Scotiabank Place (7:30 p.m., A-Channel, Team 1200).