His side took the lead after Willian put the hosts in front with his first-half strike. It looked like it would not be enough though, as Dan Gosling's injury-time curler put the Cherries back in the game with extra-time seemingly looming.
However, Morata scored just a minute later after a delightful back-heel assist from Eden Hazard to help the Blues into the last four of the competition.
As the Spaniard went to celebrate with the home faithful, he received a yellow card for his actions from referee Lee Mason in what was his fifth booking of the season.
It means Morata will now serve an immediate one-game suspension and will miss Chelsea's away trip to Goodison Park to take on a highly-rejuvenated Everton side.
"We must be pleased. We had a great reaction to their goal," Conte said following the game. "It's not simple conceding a goal in the last minute of the game, but when you have this reaction you have players with great character, personality and talent. We must be pleased.
"I'm very happy. Very happy for the performance of these players, because it means we are working very well.
"But I must be honest, I didn't see this [Morata] yellow card. He's a big loss for us, especially because we have to play a tough game against Everton. It's a pity. It's a pity. But we have to accept it. Tonight, many players were booked."
Meanwhile, Chelsea will meet Arsenal in the last four of the Carabao Cup.
Having lost to the Gunners in their last two meetings at Wembley, Conte will be looking to end a winless three-game run against the current FA Cup champions in their two-legged semi-final with the first leg taking place on 9 January 2018.