The Spaniard has scored six Premier League goals in as many games and also opened his Chelsea account in the Champions League against Real's city rivals Atletico on Wednesday.
"I came to Chelsea because one club and one coach believed in me. Now I have all of these things and I'm very happy," Morata told British media.
"Was it a big move for me to become the first-choice striker? It's the most important thing," added Morata, who struggled to make an impact at Real.
Morata said the decision to swap La Liga for the Premier League meant he was taking on a "difficult challenge" but was happy Chelsea were fighting for the league title.
"It's the most competitive league in the world. Yes, it's physical. And it's the league with the most teams who can win the title," the 24-year-old said.
"There are strong teams, but we're very happy because we're fighting for the Premier League too."
Third-placed Chelsea host early pace setters Manchester City in the league on Saturday. They have lost only one of their last seven top flight meetings with Pep Guardiola's side.
"It's another big match, another big Premier League match. It's very important for us before the international break to take the three points."
(Reporting by Shrivathsa Sridhar in Bengaluru; Editing by John O'Brien)
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