Stephan El Shaaraway put Milan ahead in the sixth minute of the quarter-final tie before Sebastian Giovinco levelled for the current Serie A leaders and defending champions who will face Lazio in the semi-finals.
Juventus, who beat Milan in last season's semi-finals, rested goalkeeper Gianluigi Buffon and playmaker Andrea Pirlo as usual in the cup while Milan were beset with injuries.
In an even match, second-half substitute M'Baye Niang nearly headed a win for Milan, who also had Riccardo Montolivo's free-kick charged down with the last kick before full-time.
Buffon's replacement Marco Storari foiled Milan at the death when he somehow managed to stop Bakaye Traore's powerful volley from 12 metres.
"It was a strong response after last Sunday and I'm pleased we got that out of our system," said Juventus coach Antonio Conte, whose side suffered a shock home defeat to Sampdoria at the weekend.
