Madrid thumped their Turkish opponents 6-0 at Santiago Bernabeu on Wednesday, with Rodrygo scoring a treble as well as providing an assist to Karim Benzema.
The 18-year-old opened his Champions League account in style – two of his goals coming inside the first seven minutes – while Benzema's double meant he reached 50 Madrid goals in the competition and Sergio Ramos added a penalty on a brilliant night for the hosts.
It was only Rodrygo's fourth start after his reported €45million move from Santos but he already has five goals to his name and feels increasingly comfortable alongside his star-studded team-mates.
"It is true that everything is going much quicker than what I expected when I arrived at Real Madrid," Rodrygo said to Movistar.
"It is a very happy night. It is a dream come true to hear the Bernabeu sing my name.
"But I have to stay calm, be relaxed and continue to work day by day."
Rodrygo appreciated a gesture from his team-mates, who signed his match ball after a memorable performance, while hailing the quality of fellow attacker Benzema, who set up his stoppage-time third goal to return the favour for the Brazilian's earlier assist.
The forward added: "Playing with Karim makes everything easier. He's an amazing player and I'm very happy to play alongside him.
"I'm sure that what they've put on the ball is very nice because my team-mates are good people and they give me lots of confidence."
Zinedine Zidane's men are now on the brink of sealing a place in the last 16 from Group A. They are comfortable in second place, five points clear of Club Brugge in third with two games to play.