Madrid have won this competition only twice since the turn of the century and coach Zinedine Zidane made six changes to the side that beat Eibar but they were never likely to be troubled by the third-tier hosts.
With the likes of Cristiano Ronaldo, Sergio Ramos and Luka Modric all missing, it was Marco Asensio who provided the inspiration at the Estadio Fernando Torres.
The visitors' momentum had slowly built throughout the first half after a slightly shaky start, but it was from a contentious penalty that the Spain international finally made the breakthrough after 63 minutes.
A series of stunning saves from ex-Madrid goalkeeper Jordi Codina looked to have kept his side in the tie ahead of the return fixture at Santiago Bernabeu, but another spot-kick moved Madrid out of sight.
Substitute Paco Candela was dismissed for picking up two cautions within seven minutes of his introduction and Lucas Vazquez netted the second goal to give Zidane's men some breathing space, their outing only blighted by debutant Jesus Vallejo receiving a straight red card for a lunge on Fuenlabrada captain Luis Milla.
Madrid started on the front foot, but they almost fell behind in calamitous circumstances as Kiko Casilla met an inswinging cross with a punch that struck Achraf and bounced just wide.
Casilla gathered Miguel Atienza's header from the consequent corner and Madrid, sensing the hosts' growing confidence, moved to respond as Dani Ceballos rifled a free-kick straight at Codina.
Madrid very nearly headed into the break with the lead as Marcos Llorente cracked a shot against the crossbar on the stroke of half-time.
Codina's touch helped that marvellous attempt on its way, but the keeper was fortunate not to see his hard work immediately undone as he spilled Borja Mayoral's low strike and was relieved to dive on the loose ball.
Just as Madrid's efforts began to wane, however, a favourable refereeing call gave Zidane's men the lift they so needed. Achraf looked to be outside the area as he ran beyond Fran Garcia and went down under a half-hearted challenge, but referee Ignacio Iglesias pointed to the spot and Asensio took full advantage.
Another unorthodox Codina stop denied Asensio a quick second, before the veteran keeper brilliantly turned Mayoral's drive over the crossbar.
And Codina was at it again when Mayoral met Achraf's cross with a firm header, the Fuenlabrada man somehow getting across to his right to make another save.
Madrid, now completely dominant, continued to push for a clinching second as Vazquez whipped a low effort wide, but it was another debatable penalty award that settled this match and potentially the tie.
Theo Hernandez grappled with Candela and was successful in his appeal for a foul, with the home player dismissed and Vazquez slotting the spot-kick under Codina, before both sides ended the game with 10 men as Vallejo's bow met a premature end after his high challenge on Milla.
