Milan beat Juve in Serie A in October and also overcame them on penalties in the Supercoppa Italiana last month.
But Paulo Dybala's volley and a Miralem Pjanic free-kick put Juve 2-0 up inside 21 minutes in a repeat of last year's final, which the Turin club won 1-0.
Carlos Bacca produced a stunning effort to halve the deficit, but Manuel Locatelli was sent off for a second bookable offence moments later.
Montella's 10 men had chances to send the tie to extra time and he was pleased with the effort displayed in the second half.
"If we played every game the way we do the last half-hour, we'd win a lot more often," he told Rai Sport.
"We have shown many times that our strength is we don't give in. I am satisfied with the lads, we're on the right track and laying the foundations at the start of this project.
"Obviously we're not at Juve's level yet, but we are on the right track. I don't think we got the wrong approach this time, but Juventus started very strong indeed, so it's not quite the same as other games."
On Locatelli's dismissal, which came after a rash sliding challenge on Dybala, Montella added: "From the bench he seemed to get the ball, but in any case he is 18 years old and will learn from this to dial back some of his vehemence in the tackle.
"I hope that I am a realist more than an optimist. We earned our place in the quarter-final, come home with a defeat, but the team is picking up a certain mentality and I think we'll have some positive lessons from this."