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.
The men from the Spanish capital will come up against their Segunda Division B opponents once again in the second leg on December 18.
Cross-city rivals and holders Atletico Madrid, meanwhile, are all but through to the round of 16 after cruising to a 4-0 victory at another Segunda Division B side - Sant Andreu.
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.
Substitute David Villa then completed the rout seven minutes from time.
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.
Recreativo Huelva did upset the odds as Ruyman Hernandez netted an 90th-minute winner to give his side a slight advantage over Levante going into the second leg.
Elsewhere, Santi Mina scored the only goal as Celta Vigo edged a 1-0 victory over Athletic Bilbao.
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