After a first half in which neither side was able to find the target, Isco looked the most likely to do so in the second period, forcing goalkeeper Francis into an impressive diving save just after the hour mark.
Carlo Ancelotti threw on Karim Benzema and Luka Modric in a bid to sway the tie in Real's favour, but was unsuccessful as Sergio Ramos squandered another goalscoring opportunity in the dying moments.
Raul Garcia set the ball rolling for La Liga's second-palced side in the 13th minute before Arda Turan scored one either side of the break to put them three up.
Girona almost pulled off one of the shock results of the round as they gave themselves a fighting chance of progression by holding top-flight Getafe to a 1-1 draw.
David Timor gave the Segunda strugglers the lead in the 15th minute before Ciprian Marica saved Getafe's blushes eight minutes before the break.
