A Cristiano Ronaldo penalty cancelled out Lautaro Martinez's opener before the Juve star scored what proved to be a 35th-minute winner in a 2-1 victory in the semi-final first leg at San Siro on Tuesday.
Ashley Young's foul on Juan Cuadrado led to the spot-kick and Ronaldo's second arrived after Inter goalkeeper Samir Handanovic came charging out, the Portuguese star tucking into an open goal after dispossessing Alessandro Bastoni.
Conte rued the mistakes made by Inter as they were left with work to do ahead of next week's second leg.
"We did it all today and our regret is enormous. Juventus didn't have to work for their goals and I don’t remember Handanovic having to make any big saves," he told Rai Sport.
"We committed two gross errors that gifted them the lead. I think the team did really well and the boys gave it their all.