Roberto Luongo had 16 saves for the Canucks.
Pekka Rinne stopped 29-of-31 shots on goal for the Predators, who saw their six-game winning streak come to an end. Mike Fisher scored the only goal for Nashville, breaking a scoreless tie at 14:01 of the second period.
The third period was all Vancouver. The Canucks scored three goals and outshot the Preds 14-2 in the period.
Burrows opened the comeback with a game-tying wrist shot off assists from Daniel and Henrik Sedin at 1:17. Burrows scored the decisive goal at 17:31 and Aaron Rome knocked in an empty-netter.