It was Suzuki's eighth of the season and third with the man advantage.
Domi earned a helper on the goal, while Kovalchuk picked up his second assist in his second game as a Hab.
The Red Wings would finally get on the board with 7:38 to go in the middle frame. With Nick Cousins in the box serving a faceoff violation penalty, Frans Nielsen scored his second of the season to make it 2-1.
After 40, Montreal held the 2-1 lead, while Detroit was leading 23-16 in shots.
Just 0:51 into the final frame, Robby Fabbri evened the score with his 11th of the season.
The Habs retook the lead at the 6:13 mark of the third. Ilya Kovalchuk made a beautiful backhand pass through the legs of a Detroit defender to Victor Mete. Mete sent the puck through the slot, where it was redirected off another Red Wing and made it past Bernier to bring the score to 3-2.