The Spain international played an important role for Los Blancos in their famous double-winning campaign of 2016-17, albeit largely from the bench.
Morata was granted permission to leave the Bernabeu in the summer when Chelsea tabled a £60m offer, however, and he has now vowed to repay Chelsea boss Conte for showing faith in him.
"Fate has given me a chance. It is true that I have made some good games, I have scored goals that have earned points," he told Le Pais. "And now I find a team that has made a huge financial commitment for a Real Madrid substitute.
"In Madrid I never played five straight games. [Chelsea] have given me everything. Conte is with me until the death."
Morata has made a positive start to his Stamford Bridge career, scoring two goals and setting up a further couple in his first three Premier League games.