James van Riemsdyk, Tyler Bozak and Joffrey Lupul also scored for the Maple Leafs, who have won four of five this season. Nashville has lost three of four.
Bernier extended his domination of Nashville, improving his career record against the Predators to 8-1. He stopped eight shots in the first period, 17 in the second and 11 in the third for his first shutout this season and the seventh of his career.
After a scoreless opening period, Kessel scored the first goal at 13:41 of the second.
Following a faceoff in the Nashville zone, the puck went below the goal line on the right side. Van Riemsdyk quickly grabbed it and whipped a backhand pass to Kessel in the low slot, where he one-timed the puck by goalie Pekka Rinne.
It was Kessel's second goal of the season. He signed a $64 million, eight-year contract extension just before the start of the season.
Just before the end of the second period, Kessel's shot hit Rinne up high and the rebound landed just to the right of the Nashville net. Rinne squatted down to cover the puck, but van Riemsdyk took the puck before Rinne could get his glove on it and carried it around to the opposite side where he slid it into the open net.
Late in the game, Nashville forward Mike Fisher was assessed a major for boarding and a game misconduct after hitting former Predators defenceman Cody Franson into the boards by the Toronto bench. Franson left a pool of blood on the ice and did not return.
Bozak and Lupul scored on the ensuing power play.
NOTES: It was just the fifth time the Maple Leafs have played in Nashville, the fewest of any NHL team. ... LW Josh Leivo and LW David Broll made their NHL debuts. Broll fought with Nashville C Matt Hendricks in the second period. ... It was the 15th anniversary of the first game in Predators history.