The teenager needed just 93 seconds to score, while Ronaldo took 62 for his strike against Deportivo Alaves in 2002.
But Rodrygo had no interest in being compared to Ronaldo, who is regarded as one of the greatest strikers of all-time.
"I don't want any comparisons with Ronaldo being made because he's one of the game's all-time best and one of my country's best," he told reporters.
"As for the goal… I controlled it well and knew that from there on in things would work out. I have to always show that I'm capable of playing here."
Rodrygo's goal sealed Madrid's win after a 20-yard stunner from Vinicius Junior had put them ahead at the Santiago Bernabeu.
After a goal on debut, Rodrygo said it was a dream come true, while also praising coach Zinedine Zidane.
"The gaffer knows what he's doing and he's been talking to me every day to tell me to keep working hard," the former Santos forward said.
"I'm at Real Madrid and I don't care whether I play for Castilla, under Raul, or for the first team. Madrid are always going to have great players, so all of this prestige is to be expected.
"I'm so happy. Zidane transmits a real calmness and a lot of trust and I think that's why things went so well.
1 - Rodrygo Goes is the fastest player to score his first LaLiga goal for Real Madrid (93 seconds) since Ronaldo Nazário in 2002 (62). Impact. pic.twitter.com/EknwWdo0oi— OptaJose (@OptaJose) September 25, 2019
"I'm speechless, I've been dreaming of this moment since I joined Real Madrid and running out at the Bernabeu with the fans chanting my name… I'm so happy."
Meanwhile, Vinicius was also delighted to get on the scoresheet, netting for Madrid for the first time since February.
"The players know that I was trying hard and there's nothing better than getting back amongst the goals before a derby," he said, via the club's website. "It's an honour to speak with Roberto Carlos and I'm happy because he congratulated me. I'm very happy to score again at the Bernabeu.
"I'm excited because since my injury it's been difficult, I was finding it hard playing again and I wasn't doing well in games. I wasn't as happy as last season and it's a massive weight off my shoulders. I'm very happy when I'm scoring goals."
A point clear atop the LaLiga table, Madrid face Atletico in the derby on Saturday.