Modric came on as a second-half substitute as Madrid beat Barcelona 2-1 on Wednesday, and the Croatian has admitted that it was a dream start.
He said: "I feel great. I couldn't ask for a better beginning for my career at Real Madrid.
"I am very happy and I would like to thank the supporters for the great support during the whole match and for a great reception they gave me when I came on."
Madrid go in search of their first La Liga win of the season when they play Granada on Sunday, September 2.