The Brazilian, 18, opened the scoring at Santiago Bernabeu just four minutes into Madrid's Group A clash to mark his first-ever goal in the competition.
And he would not have to wait long to double his tally.
Two minutes later Rodrygo found himself in the right place to head Marcelo's cross past Fernando Muslera, the left-back serving up both strikes for his young compatriot as Madrid took a commanding lead.
In total the former Santos prodigy needed a mere six minutes and 14 seconds to net twice past the Turkish visitors, the fastest double ever recorded in the Champions League.
Any hopes he may have had for a hat-trick inside the first quarter-hour, however, were to be dashed.
2 - Real Madrid's Rodrygo Goes has scored the earliest brace in the Champions League history (6' 14''). Thunder. pic.twitter.com/qQglFKmH3i— OptaJose (@OptaJose) November 6, 2019
When Toni Kroos was brought down in the area it was captain Sergio Ramos who stepped up to do the honours from the penalty spot, and he obliged with a panenka finish to put his side 3-0 up.
Had Rodrygo been given the chance to score from the spot and beaten Muslera once again, he would have broken the record for the fastest hat-trick in the competition's history, beating Marco Simeone's 24-minute treble for AC Milan against Rosenborg in 1996.
However, in the second minute of stoppage-time, Rodrygo converted Karim Benzema's pass to complete a perfect hat-trick by scoring a treble using his right foot, left foot and head.
In doing so he became the youngest player to score a perfect hat-trick in the competition at 18 years 301 days, smashing the record set by Paris Saint-Germain striker Kylian Mbappe, who was 20 years and 306 days old when he hit his treble against Club Brugge in October.
He is also the second-youngest player to score a hat-trick in the Champions League behind only Real legend Raul, who was 18 years and 113 days when he scored his treble for Los Blancos against against Ferencvaros in October 1995.
Benzema scored a late double himself to secure a thumping 6-0 victory for Zinedine Zidane's side.
The glut of goals will be more than welcome for Madrid and Zidane, particularly following their Liga toils at the weekend.
"We won’t score five goals every day,” Zidane told reporters after Saturday's stalemate.
"We have had chances. The goal is the only thing that we have lacked."
Fortunately for the Frenchman they were in no short supply in the Champions League, thanks to their dangerous teenage frontman.