The Spain international came off the bench to replace Michy Batshuayi with 19 minutes to go in the Carabao Cup tie after Willian broke the deadlock in the first half at Stamford Bridge.
However, former Newcastle United midfielder Dan Gosling levelled in the 90th minute to nudge the League Cup tie towards extra-time.
But summer signing Morata netted a dramatic winner deep into injury time in west London to set up a semi-final meeting with Arsenal for a place in the final at Wembley at the end of February.
Chelsea handed a rare start to former Manchester City goalkeeper Caballero for the cup clash and the Uruguay international made an impressive save to stop Lyn Mousset.
While Morata was heralded as the match winner at Stamford Bridge, the Chelsea striker was quick to deflect praise on the Blues shot stopper.
The Chelsea striker wrote on his Instagram page:
“What a goalkeeper! what a person! and what a great friend! You’re a phenomenal”
The Blues signed Morata in a £58m deal from La Liga giants Real Madrid in the summer transfer window.
MORE: Man United latest news
MORE: Arsenal latest news
MORE: Chelsea latest news
MORE: Liverpool latest news