For the second year in a row, Juve needed extra time to edge past the Rossoneri, and they began on the back foot as Stephan El Shaarawy fired Milan ahead in the sixth minute.
But Sebastian Giovinco squeezed in a classy free-kick to force 30 more minutes of play in Turin, leaving substitute Mirko Vucinic to score five minutes into extra time and send Juve through to the last four.
Conte was especially delighted with his team's response in the wake of a disappointing home defeat to Sampdoria in Serie A.
He told juventus.com: "The guys knew that what happened on Sunday could never happen again, not so much the result but the performance.
"And I saw a great response from all of them, especially those who have had less chances to show what they can do. Milan are a great team and they turned up with their best side. We wanted to get through this round and we succeeded."
Juventus will now face Lazio in a two-legged semi-final, with Milan boss Massimiliano Allegri left pondering what might have been.
"I'm happy about the players' performance," he told acmilan.com. "But we missed a few chances in the second half and it's a shame.
"We had a few chances to equalise but we couldn't do it. We ended the game with four players on the field who are 20 years old. The road that the club is on is the right one but time is needed. (Kevin-Prince) Boateng played a good game, he has always played well when he plays out wide and he made very few mistakes.
"(Bakaye) Traore came on well just as Bojan and (M'Baye) Niang did. The team is growing and now we have to think about Sampdoria as that's an important game for us."