The Portugal international hit his ninth goal of Real's Group B campaign in the 2-0 victory over Copenhagen in Denmark on Tuesday.
His second-half strike lifted him above former joint record holders Ruud van Nistelrooy, Filippo Inzaghi and Zlatan Ibrahimovic, with the Swede having also scored eight times before being left out of Paris Saint-Germain's final group game.
Ronaldo's achievement is made all the more impressive by the fact he missed Real's penultimate Champions League clash against Galatasaray with injury.
He said: "I'm happy with the record. I have scored another goal and I have helped the team. I'm happy with that. It is just a record, though, and to be honest I don't worry too much about that."
The 28-year-old, who missed a late penalty, had been suffering with a hamstring injury and claims he has not yet returned to full fitness.
"As I have been sidelined for a couple of weeks, I have to get back into the rhythm," he added. "I feel good, but I'm not at my best physically.
"I am really enjoying my football at the moment. My own play is more fluent, and I have to continue like this, but I know that the season is very long.
"I have not felt good in all matches, but today I played well, and I want to enjoy that. We are playing well as a team right now. We play some good football, attractive football, and have a lot of confidence."
Coach Carlo Ancelotti paid tribute to Ronaldo's ability, describing him as a "complete forward".
"I think that Inzaghi will not be happy tonight because he has lost this record," the Italian said. "He has gone ahead of a very strong player. Inzaghi was very strong in the area and Ibrahimovic is powerful and has a fantastic ability to score and assist.
"Cristiano Ronaldo is a complete forward who can score goals in every way possible. He's a bit angry about the penalty, but he scored an important goal and played very well."
