Madrid progressed to the last 16 with a 4-2 aggregate win but Zidane's men were forced to dig deep against the unheralded visitors in the second-leg fixture at the Santiago Bernabeu on Tuesday.
Fuenlabrada took a deserved 25th-minute lead thanks to Luis Milla's stunner in the Spanish capital – where Gareth Bale returned from injury – before a quick-fire brace from Borja Mayoral in the second half allayed fears around the stadium.
Alvaro Portilla struck at the death to secure a memorable draw for Fuenlabrada but head coach Zidane, who named the youngest team of his Madrid reign, was upbeat post-match.
"I'm not upset at all, it was foreseen," Zidane told reporters. "It's not easy for the players, there were homegrown players.
"I'm happy to advance in the tie, the opponent played a very good game. The important thing is the tie and we passed."
Zidane added: "People can say what they want, but they also saw their players fight until the end.
"There were players who had not played together. The opponent does not have a level of Segunda B. Fuenlabrada are a good team.
"I'm looking at my players, those who were on the field, in the second half we did it well, I'm happy,."