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.
Ohio bar shooting arrested, charged with murder
LGBT community has 'bullied the American people': Bachmann