Carlo Ancelotti's side kept up their record of having kept clean sheets in every Copa game so far this season as they look to atone for last season's final defeat to Atletico Madrid.
The shut-out was also their eighth consecutive since the turn of the year as they once again displayed the full extent of trophy credentials.
Real's goal came in the seventh minute courtesy of Jese Rodriguez, who raced clear after a pin-point Xabi Alonso throughball before firing past Kiko Casilla - who could have done better at his near post.
Cristiano Ronaldo's inclusion was perhaps a surprise but the 2013 FIFA Ballon d'Or winner looked determined to make his mark on the game, Casilla palming away a chip from the Portuguese and denying him when one-on-one in quick succession.
Casilla, a Real academy graduate, was kept busy in the first half, acrobatically saving Angel di Maria's deflected shot shortly before the interval.
At the other end, Real captain Iker Casillas had very little to do but showed his class when substitute Jhon Cordoba was sent clear 10 minutes into the second half, narrowing the angles well to deny the Colombian.
The second half saw few chances as Real tightened their grip on the game, and the visitors' faint hopes were all but extinguished when Victor Sanchez was dismissed for a second bookable offence.
Despite that setback, Raul Rodriguez wasted a gilt-edged chance to reduce the overall deficit, turning his shot into the side netting from a deep free-kick.
Real could face their cross-town rivals Atletico if they overcome Athletic Bilbao on Wednesday, with the visitors taking a 1-0 lead to San Mames.
