During the summer, Morata was brought in as a replacement for Costa, who opted to leave Stamford Bridge for former club Atletico Madrid after falling out of favour with head coach Antonio Conte.
Morata has acknowledged that it was "a big responsibility" to replace Costa - who netted 59 goals in 120 appearances for the Blues - but has suggested that his fellow Spanish international has helped him settle into life in England.
The 25-year-old told Sky Sports News: "He is a very good player and a good man too. I'm Diego's friend and when I came here I knew that is was a big responsibility to replace him, but I've tried to do well.
"Always when we are together in the Spanish team, we talk about football and life. We talk about everything and I always say thanks to Diego."
Morata has scored 12 goals in 24 matches in all competitions this season.