Eighteen-year-old Rodrygo netted twice in the first seven minutes at Santiago Bernabeu and later added a stoppage-time third as he became the youngest Brazilian to score in Europe's premier club competition.
Sergio Ramos' penalty had Madrid three up with the game effectively over after 14 minutes and Benzema scored in each half to reach 50 goals in the competition for Madrid, making him only the fourth player to achieve the feat for a single club.
After a poor start to their Group A campaign, Madrid are now five points clear of third-placed Club Brugge – who lost to Paris Saint-Germain on Wednesday - with two games to play, while Gala are out as a result of their chastening defeat.
Rodrygo's opener was impressive as he collected Marcelo's cross before cutting inside past Mario Lemina to send a fine left-footed finish into the far corner.
He doubled his tally when a mistake from Mariano gave Marcelo the chance to set up his compatriot again, the forward this time heading in a left-wing delivery from eight yards.
VAR ruled in Madrid's favour when Toni Kroos – who scored the only goal in the reverse fixture in Turkey a fortnight ago - complained he had been fouled by Steven Nzonzi right on the 18-yard line, Ramos beating Fernando Muslera with a cheeky Panenka from the resulting spot-kick.
Casemiro had a goal ruled out and Eden Hazard was denied by Muslera, but Benzema ensured it was four in the last minute of the first half with a simple finish, an assist from the impressive Rodrygo coming after a dreadful pass from Yuto Nagatomo put Gala in trouble.
After the break, Muslera denied Kroos before Christian Luyindama blocked Benzema's follow-up near the goal-line, with the goalkeeper again called into action to keep out substitute Isco from the resulting corner.
Thibaut Courtois thwarted Lemina at the other end, but Benzema struck again after 81 minutes, converting Dani Carvajal's cross from close range.
And Benzema turned provider two minutes into added time, his throughball cleverly finished by man-of-the-moment Rodrygo.
What does it mean? Madrid well placed after slow start
Madrid missed the chance to go top of LaLiga at the weekend with a poor home draw against Real Betis but after winning just four of their first nine games to start 2019-20, they would probably have taken being level on points with Barcelona and on the brink of a last-16 spot in Europe by early November. Rodrygo gives the Madrid fans something to get excited about, with a possible injury to Marcelo, who was taken off by Zinedine Zidane just before the interval, the only concern.
Rampant Rodrygo making his mark
It was only his fourth Madrid start, but Rodrygo already looks the part having become the second-youngest scorer of a Champions League hat-trick, behind Madrid legend Raul. The ex-Santos star took all three of his goals clinically and showed his creative ability with a clever delivery to set up Benzema's first goal. It was also a fine night for Benzema, whose strikes ended his four-match streak without scoring in Europe to reach the 50-goal milestone that had only previously been achieved by elite company in Lionel Messi, Cristiano Ronaldo and Raul.
15 - Karim Benzema is the second player to score in 15 consecutive Champions League seasons after Messi. Legend. pic.twitter.com/0Xvqv6N8Cg— OptaJose (@OptaJose) November 6, 2019
Key errors cost woeful Gala
Gala are winless in 16 Champions League away games and the best they can now hope for is a Europa League spot. They are the first team in nine years to fail to score in six straight away games in the competition. While the latter stat is on their attacking and creative players, it was defensive mistakes which cost them dearly here, with Mariano, Nzonzi and Nagatomo all guilty of dreadful first-half errors.
Key Opta Facts
- This is just the second time Galatasaray have conceded at least six goals in a Champions League match, also doing so against Madrid back in September 2013.
- Gala have now conceded 27 goals against Madrid in the Champions League – 13 more than against any other side in the competition.
- Benzema has now scored in 15 different Champions League campaigns – only Ryan Giggs has scored in more (16).
- Benzema scored his 50th goal for Madrid in the Champions League, the fourth player to do so for a single side after Lionel Messi (113 for Barcelona), Cristiano Ronaldo (105 for Madrid) and Raul (66 for Madrid).
Madrid are at Eibar in LaLiga on Saturday, with a home game against Paris Saint-Germain next up in this competition on November 26. Gala are away to Gaziantep on domestic business before hosting Club Brugge next in Europe in a key match in the battle to finish third.