The 21-year-old showed a glimpse of his father's talent with a fine low finish from the edge of the area just after the hour mark.
By contrast, Barcelona still have work to do to reach the next round after being held 1-1 at Hercules in the first leg of their last 32 tie.
Enzo, the oldest of Zidane's four sons, all of whom play at different levels in Real's youth teams, was introduced by his father, a legendary former player for Real and France, at half-time.
The 21-year-old, who was previously coached by Zinedine at Real's youth side Castilla before he took the top job at the Santiago Bernabeu in January, showed a glimpse of his father's talent with a fine low finish from the edge of the area just after the hour mark.
"If I take off my hat as a coach then I am happy for him as a father (too)," said Zidane.
"But, in saying that, I watch what he does on the field as a coach and I am happy with how they all played."
With the tie over as a contest following Real's 7-1 first leg win and El Clasico away to Barcelona to come on Saturday, Zidane made nine changes from the side that started the 2-1 win over Sporting Gijon last weekend.
Norwegian starlet Martin Odegaard, 17, was also handed his full debut, but it was Dominican international Mariano Diaz who got Real off to a flying start as he slotted home the first of his hat-trick after just 23 seconds.
James Rodriguez headed home the hosts' second before Mariano smashed in a third, but Leonesa did have a moment to savour when Yeray Gonzalez fired into the top corner from long range in first-half stoppage time.
Enzo was then handed his long-awaited bow by his father, but ensured there was to be no claims of nepotism when he finished off a flowing team move with a classy right-footed finish.
Mariano completed his hat-trick two minutes from time before a Cesar Morgado own goal.
Barcelona also fielded a side with 11 changes ahead of the Clasico, but were made to pay by third tier Hercules.
David Mainz bundled home at the back post seven minutes into the second-half to give the side form Alicante a shock lead.
However, Manchester United target Carles Alena capped his debut with a wonderful strike from 25 yards to spare Barca's blushes.
Yannick Carrasco scored twice as Atletico Madrid also hit Guijelo for six in the first leg of their last 32 tie.
A Saul Niguez penalty and Sime Vrsaljko's first Atletico goal gave the visitors a 2-0 half-time lead before the Belgian struck twice in five second-half minutes.
Angel Correa and Roberto Nunez rounded off the scoring.
Sevilla and Villarreal took huge strides towards the last 16 with comprehensive first leg wins over Formentera (5-1) and Toledo (3-0) respectively.
Second division Cordoba were the only victorious lower league side of the evening with a 2-0 win over Malaga.