Rafeal Leao caused problems for Inter in a match where chances were at a premium, but the Rossoneri were unable to make their superiority count against their out-of-form city rivals.
With Italy's domestic cup not following UEFA's example of removing the away goals rule, however, the stalemate will certainly be more kindly looked upon on the red side of Milan, who are level on points with Napoli at the top and two ahead of Inter in a fierce race for the Scudetto.
Alexis Saelemaekers tested Samir Handanovic during a frantic start after the stadium observed a heartfelt message from Milan's Ukrainian legend Andriy Shevchenko, who scored in their last Coppa Italia final win in 2003.
Milan's Theo Hernandez shot wide after a trademark burst forward, while Alessio Romagnoli denied Inter with a superb defensive header on 20 minutes before limping out of the contest injured.
Stefano Pioli's men enjoyed the better of the first period, with Rade Krunic shooting over from a great position and the lively Leao troubling Inter in an otherwise uninspiring opening 45 minutes.