Willian scored to give Chelsea an early lead, and it looked like it would be enough as Bournemouth struggled for clear cut chances.
Until Dan Gosling popped up on the 90th minute mark to curl a sublime strike past Willy Caballero and send the Cherries fans into delirium.
Their joy lasted less than a minute though when Eden Hazard set up Morata with a tap-in straight after the kick-off.
17-year-old defender Ethan Ampadu was a standout for Antonio Conte's side, with Sky Sports labelling him the man-of-the-match.
Speaking after the game, Conte said: "The most important thing is that we are in the semi-finals.
"Last season we were eliminated in the fourth round by West Ham.
"This season we must be happy with this achievement and to reach the semi-final of one of the two cups in England. It's great for us, especially because we reached the semi-final with a lot of rotations.
"From the first game against Nottingham Forest I made a lot of rotations, and for this reason I am very happy. I had good responses from my players. In this cup they showed they are very professional.
"I hope this game also improves the physical condition especially for the players who had injuries and who are not playing with regularity. It means we are working very well.
"To reach the semi-finals is a remarkable achievement for us. We want to reach Wembley and play the final, but we know the other teams want the same."