Brassard put the Senators ahead 1-0 at 15:12 of the first period. He tipped Holden's shot from the point inside the left post after Mark Kastelic won an offensive zone face-off.
Matheson tied it 1-1 at 4:29 of the second period when he skated around Ottawa defenseman Thomas Chabot and flipped his own rebound over Sogaard.
Suzuki put the Canadiens ahead 2-1 at 8:46, deking Sogaard on a breakaway for his 20th goal of the season.
Shane Pinto tied it 2-2 at 13:51 when he redirected Alex DeBrincat's pass in the slot.
"We didn't take care of the puck at all," Senators coach D.J. Smith said. "Clearly we were rattled there for a six-minute span where we were just kind of running around, turning pucks over, lost focus really. A game like this, we knew we had to have it, and sometimes you squeeze and it's not going your way.
"But good on the guys, they got themselves together after the second TV timeout, played much better in the second half."
NOTES: Sogaard, who has appeared in five games this season, is 3-0-1 in four starts. … Ottawa has won six straight against Montreal for the first time since Feb. 8-Nov. 19, 2007. … Kastelic had an assist and two hits in 11:46 in his first game since Jan. 21 after being recalled from Belleville of the American Hockey League on Saturday. … Matheson has five points (two goals, three assists) during a four-game point streak.