A much-changed Chelsea side took the early lead through Willian but Bournemouth midfielder Dan Gosling levelled the score at 1-1 in the 90th minute.
The hosts got the winner through an Alvaro Morata goal less than a minute later and Conte was thrilled to see Chelsea progress to the semi-finals, where they will meet London rivals Arsenal.
”The most important thing is we’re in the semi-finals,“ Conte said after the match. ”To reach the semi-final in one of the two cups in England is great for us, especially with a lot (of squad) rotation.
”I had good responses from my players and I hope to continue in this way. Many players had their chances tonight to play from the start and I think I had good responses.
“It’s also not simple when you concede a goal in the last minute of the game, so to have this type of reaction shows great character, personality and heart.”
Premier League champions Chelsea, who are currently third in the standings, will travel to resurgent Everton on Saturday.