Torino were reduced to 10 men in the 69th minute when defender Koffi Djidji received a second yellow card for a challenge on Milan substitute Junior Messias.
With the game looking certain to go to penalties, Adopo wrapped up victory for Torino in the 114th minute with a close-range strike into the bottom corner to stun the hosts.
Torino, who have won the Italian Cup five times but not since 1993, joined Milan's local rivals Inter in the last eight after Simone Inzaghi's side advanced with a 2-1 comeback win over Parma on Tuesday.
