Mbappe hit a hat-trick as Real won 3-1 to win their round of 16 playoff 6-3 on aggregate.
He said at the final whistle: “It was a perfect night for the team and we wanted to win to reach the last 16. When you play against Manchester City it's always a tough game. We're strong at home, we played collectively at our best and we gave joy to all the Madridistas.
“It was a great day for us and for the team. We have to keep this up if we want to win the things we want to win.
“I didn't want to come here to perform badly. I want to play well, to mark an era and write history at Real Madrid. The adaptation period is over and I have to play with personality. I have no limits, but the most important thing is to win trophies. I've scored goals in my career, but I haven't won many trophies. It is in my blood to score a lot of goals and win many trophies.”
On their next opponents in the competition, Mbappe added: “Atlético and Bayer Leverkusen are two top teams and very difficult opponents. For me, it's better for us to play Atletico because then we don't have to travel. Both will be difficult, but Atlético would be tough without travelling.”