Desharnais checked Mattias Ekholm into the boards, sending the Predators defenseman to the ice on what Nashville claimed was a trip. The puck came loose and Desharnais picked up it, skated into the left circle and ripped a high shot past Carter Hutton for the game-winner.
The overtime score came after Montreal had surrendered a 3-0 lead.
The Predators' Shea Weber had forced overtime with a power-play goal with 6:20 remaining in regulation.
Nick Spaling and Gabriel Bourque also scored.
Marek Mazanec started in goal for Nashville but was lifted in the first period after yielding three goals on five shots.
Hutton stopped 17-of-18 the rest of the way.