Kessel's third tally, with 2:19 to play, gave the Leafs a two-goal cushion, which they ended up needing when Ottawa rookie Stephane Da Costa recorded his first career goal 25 seconds later.
Joffrey Lupul added a goal and two assists as Toronto had to hang on after piling up a 5-1 lead through two periods.
James Reimer made 32 saves, but allowed three markers in the third period, when Ottawa launched 16 shots.
The Senators fell to 0-2-0 despite two goals from Daniel Alfredsson.
Notable deaths of 2014 [PHOTOS]