Kesler, who had two goals in the contest, was perfectly positioned in front of the net at 10:45 of the overtime period and tipped a slap shot by Mikael Samuelsson past Pekka Rinne to give the Canucks a 2-1 lead in the series.
Kesler scored his first goal at 1:00 of the second period to tie the score 1-1. Chris Higgins added the other Vancouver tally in the third period.
Higgins' goal at 3:03 put Vancouver in front 2-1 but the Predators tied the score at 13:18 and sent the game into overtimes. Nashville led 1-0 after one period on a shorthanded goal by David Legwand.
Roberto Luongo stopped 28-of-30 shots in goal for the Canucks. Rinne faced 47 shots.
The series resumes Thursday in Nashville.