It remains to be seen whether Enzo will manage to live up to the astounding achievements of his father, who won one LaLiga title and the UEFA Champions League during his five years as a player at Madrid.
But the 21-year-old made an impressive start at the Santiago Bernabeu, scoring with his first shot less than 18 minutes into his first appearance in Madrid colours.
Forward Mariano scored a hat-trick on his first start as Madrid secured a place in the last 16 courtesy of a 13-2 aggregate victory.
Martin Odegaard and Alvaro Tejero also made their full debuts, and Zidane conceded it was a possibility some of the younger players may be moved on to experience regular first-team football.
"I am proud as a father and manager for Enzo's goal," Zidane said. "Not for just that but for the entire team and how they played tonight.
"We achieved what we wanted and played very strongly while also giving the youngsters minutes and some their debuts.
"If I was to take my manager's suit off I'd say I am very happy for my son. As a coach I am very happy for all the players. Mariano's goals were great. Tonight has been a great night for all the fans.
"The idea was to give Mariano minutes. I told him in pre-season to stay with us and we'll see what happens in the winter market. As for now, Mariano is our player and he seized his minutes very well.
"I do not know what will happen to Odegaard, we will see later. At 17 he has character and personality. He's done very well."
Midfielder Casemiro played the full 90 minutes on his first appearance since fracturing a leg in mid-September and, with a Clasico trip to take on Barcelona at Camp Nou just three days away, Zidane was glad to have the Brazilian back in action.
"Casemiro got better and better as the game went on. It was important for him to play a full game, and he is physically ready now," Zidane said.
"If you saw our last few games, we were doing well in midfield. We'll see later who plays on the weekend. Now we just rest because our next game is just three days away. Casemiro was very strong but played after two months. We'll see.
"We all need some well-earned rest now. From tomorrow, we'll start preparing for El Clasico."