Michael Pezzetta and Cedric Paquette picked up assists on the tally.
Brayden Point, returning from a 14-game absence, followed with a pair of unanswered goals to give the Lightning a 2-1 lead.
Tampa carried that lead into the room after 20 minutes, and was also leading 10-9 in shots.
The Habs got back in the game early in the second, with Harvey-Pinard celebrating his first NHL game with his first goal in the big show. The Jonquiere native waited patiently in the slot and one-timed a nifty backhand pass from Sami Niku past Lightning goalie Maxime Lagace.