The Argentina international sustained the injury just seven minutes into the 2-0 victory at the San Siro and is now set for a long spell on the sidelines.
An Inter statement read: "Diego Milito underwent diagnostic tests at the Fondazione IRCCS Policlinico San Matteo in Pavia this evening, after being forced off the pitch with an injury in the seventh minute of the match against Cluj.
"The tests revealed that the striker has damaged his anterior cruciate ligament and fibula collateral ligament."
It is another blow for Milito, who returned from a month-long absence in Sunday's Serie A win over Chievo.
Inter boss Andrea Stramaccioni admitted the injury was the major downside of a win achieved thanks to two goals from Milito's replacement, Rodrigo Palacio.
Stramaccioni told Mediaset: "Unfortunately it takes some of the shine off tonight's [Thursday's] win, which was otherwise very positive.
"It's his cruciate ligament, a new type of injury for Diego, nothing to do with the problems he's had before.
"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. We have to wait for more information."
Milito's absence is a major blow to fourth-placed Inter's hopes of qualifying for next season's Champions League.
Stramaccioni added: "Milito is a huge loss for us because he's our only out-and-out centre forward.
"We'd planned for him and Palacio to play part of the game each tonight [Thursday] to keep them both fresh, but when you get injured seven minutes into the match it's just sheer bad luck.
"We're all saddened, all of us, and I feel particularly close to the lad personally. We're all very down about it."