The Argentine, who scored both goals in the final when Inter won the Champions League in 2010, had only just returned from injury and was in tears when he was carried off after six minutes at the San Siro.
"It's his cruciate ligament, a new type of injury for Diego, nothing to do with the problems he's had before," coach Andrea Stramaccioni said.
"We're gutted for Diego as a person first and foremost, and then of course for what he represents as a player. We need to see when he'll have the operation."
Compatriot Rodrigo Palacio replaced Milito and scored both goals in the first leg of the round of 32 tie but he is generally a winger.
Inter, fourth in Serie A, must now face the final three months of the season without their only real out-and-out striker.
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- Andrea Stramaccioni