Matt Stajan contributed three assists and Vesa Toskala made 23 saves in Toronto's second victory in three games.
Kaberle's one-timer 2:47 into the second period tied the game and the Maple Leafs took the lead on a power-play goal from Nikolai Kulemin less than two minutes later.
After Morneau lifted the Oilers into a tie, Toronto went in front for good when Nik Antropov scored from in front with 61 seconds remaining in the second.
Notable deaths of 2014 [PHOTOS]
LeBron James reveals his new Nike shoes