Caufield won it with a one-timer from between the circles off a backhand pass from Nick Suzuki.
It was the fifth goal in three games and second straight overtime goal for Caufield, who scored at 3:25 to give the Canadiens at 5-4 win against the Seattle Kraken on Tuesday.
Caufield drew Montreal even 2-2 with 20 seconds remaining in the third period and goalie Jakub Dobes pulled for an extra attacker. He snapped a shot past Saros from the left face-off circle after taking a long pass from Lane Hutson, who had just stopped Jonathan Marchessault’s shot that was headed toward an empty net.
Oliver Kapanen scored, Hutson had two assists, and Dobes made 17 saves for Montreal (4-1-0), which got its fourth straight win.
Steven Stamkos and Nick Perbix scored, and Juuse Saros made 27 saves for Nashville (2-1-2), which has lost consecutive games (0-1-1).
Stamkos gave the Predators a 1-0 lead with a power-play goal at 11:36 of the second period. His slap pass toward Michael Bunting at the edge of the crease struck Canadiens defenseman Kaiden Guhle’s right skate and deflected past Dobes’ glove into the net.
Kapanen tied it 1-1 at 6:02 of the third when he put his own rebound between Saros’ pads.
Perbix put Nashville up 2-1 at 11:21 with a wrist shot past Dobes’ right pad from the right face-off circle, eight seconds after Saros topped Mike Matheson with the pad at the left post.