It was Drouin's fourth goal of the year.
Suzuki also collected an assist to reach the 100-point plateau for his NHL career.
Montreal led 2-1 after 40 minutes.
The Lightning ultimately secured the victory with some late-game heroics.
After Corey Perry scored the equalizer with 2:09 left in regulation, Ondrej Palat netted the game-winner with 38 seconds to go.
Allen finished the contest with 24 saves on 27 shots.
The Canadiens are back in action on Thursday night when they host the Chicago Blackhawks at the Bell Centre.