Gustav Lindstrom tied it 4-4 for Montreal with 3:59 remaining in the third when he scored against his former team on a shot from the right side on a pass from Josh Anderson.
Suzuki scored 4:12 into the third to get Montreal within 4-3 with a wrist shot from the right circle.
"The third period was good, but we can't start the game that slow because it's hard to come back from 3-0," Lindstrom said. "The guys did a good job battling back in the third but we have to start the game better."
Joe Veleno put Detroit up 1-0 at 7:21 of the first period taking Moritz Seider's pass from the point and scoring from the top of the right circle.
Christian Fischer made it 2-0 with a short-handed goal at 11:44. He tapped in the puck as it lay in the crease to the left of Allen, who lost sight of it after stopping Michael Rasmussen on a breakaway.
"I made a breakaway save, he was going blocker side and there was a big man coming down on me, and I don't really know what the heck happened after that, to be honest," Allen said. "I had no idea where the puck was."