The Portuguese forward grabbed three second-half goals as Real romped to a 6-1 away win in their Champions League Group B opener on Tuesday, taking his tally with the nine-times European champions to 206 goals from 204 games.
Bale's arrival from Tottenham Hotspur for 100 million euros this month, surpassing the then-record 94 million they paid for Ronaldo in 2009, prompted the club to move swiftly to keep their leading scorer happy.
Ronaldo agreed a new contract with Real president Florentino Perez on Sunday which ties him until 2018 and reportedly makes him the world's highest-paid player with a net annual wage of some 17 million euros.
"The last few days have been great for me," Ronaldo told reporters.
"I am very pleased. We did what we had to do, which was to win, and the others drew (Copenhagen held Juventus 1-1), which is even better.
"Our next two games are at home so if we win them we will be doing really well towards making the next stage."
Bale came on as a second-half substitute and played a part in setting up the fourth and fifth goals for Ronaldo and Karim Benzema.
Ronaldo, who spent six years in England with Manchester United, was the first to greet Bale when he arrived for his debut training session last week, seemingly taking him under his wing and speaking to him in English, and has been seen pushing him hard in training.
"We are getting to know each other," Ronaldo said. "He is steadily getting up to speed because he has been out of action for a while and it is all going well for him."
Bale joked at his official presentation that Ronaldo was the boss when it came to team hierarchy and the taking of free kicks, but the midfielder got to take the set piece that led to Ronaldo's second goal.
"It was great to be back in the Champions League again," Bale, who scored on his league debut against Villarreal at the weekend, told reporters.
"It was nice to be able to contribute immediately, taking the free kick. It was fantastic to be involved in all three of the goals at the end.
"It was important to be able to show what we can do. It was great to play for a bit to help me improve my physical fitness."
Fellow new recruit Isco has also set a high standard for Bale to match. He scored the all-important opening goal against Galatasaray, to make it four goals in five appearances in all competitions so far.
France striker Karim Benzema's brace helped him to respond to rumblings of discontent with some of his recent performances, in particular with his national team.
"I'm feeling good. I had a difficult moment (being a substitute for France's last international outing) but I am looking to change this quickly," the 25-year-old told reporters.
"It's not a problem. I feel very settled at the club, happy, and I work hard to put in these types of performances."