Kessel's tally early in the second period proved to be the game-winner, putting the Leafs ahead for good at 3-2.
He later added an empty-netter to tie Pittsburgh's Evgeni Malkin for second place on the NHL's goal-scoring list.
James Reimer came off two consecutive shutouts to turn aside 27-of-30 shots for the win.
Jordan Eberle scored twice and Jeff Petry added a goal and a helper for Edmonton.
Devan Dubnyk absorbed the loss despite making 24 saves.
2014: The Year in Music [PHOTOS]