His headed goal was enough to see Chelsea through to the last four where they will play Southampton.
As it stands, Manchester United, Tottenham Hotspur, Southampton and Chelsea will be looking to secure the two spots of the final.
Chelsea were made to work hard to progress in the only remaining competition it can win this season.
"For sure, for the striker is very important to score", said Conte. Willian won the ball near his own penalty area and found Morata with a fine pass. Morata took one touch and then sent the ball over the outrushing Kasper Schmeichel with a fine finish.
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Spanish forward, Alvaro Morata had slid Chelsea ahead just before half-time, but his effort was cancelled out by a Jamie Vardy equaliser on 76 minutes, which forced the game to extra time.
Substitute Pedro leapt the highest to find the back of the net after a sublime delivery from N'Golo Kante coupled with an erratic decision to come out of his line from Kasper Schmeichel gifted Chelsea with the ideal escape route out of East Midlands.
But the visitors improved in the second half and Pierre-Emile Hojbjerg made the breakthrough in the 62nd minute when he converted a corner at the near post.
However, the England worldwide made amends 15 minutes from time when he forced the ball home at the third attempt after Vicente Iborra's first two efforts were blocked and then saved by Willy Caballero.