Osasuna held Barcelona and Real to draws at El Sadar in the league this season but the visitors, fired by the confidence of a 2-0 first-leg win, had far too much for them in attack.
Ronaldo, named FIFA World Player of the Year on Monday, was fortunate to score, however, as his free kick slipped through the hands of keeper Andres Fernandez after 20 minutes.
Angel Di Maria slotted in the second to seal a comfortable win even though Real defender Fabio Coentrao was given his second yellow card six minutes from time.
"We knew it was going to be difficult and that Osasuna were going to come out strong, so it was good to get the opening goal. The shots from Cristiano are difficult to stop as they move around a lot," Real midfielder Asier Illarramendi told reporters.
"We worked hard but now we need to keep going and next we have Betis in the league at the weekend.*
"Real put out a strong side but Karim Benzema and Gareth Bale were left out of the attack with Isco given the unfamiliar role of leading the line."
As Real began to relax towards the end of the first half, Osasuna threatened and Alvaro Cejudo hit the woodwork with a 25-yard drive while Miguel De las Cuevas forced a smart save from Iker Casillas.
Their heads dropped, though, when Di Maria fired home a Jese Rodriguez cross 10 minutes into the second half. With the match all but won Coentrao needlessly fouled Roberto Torres and was given a second booking but Real still sealed an easy win.
Espanyol survived a scare as they went a goal down having lost the first leg 1-0 at second division Alcorcon but they responded with a 4-2 win and 4-3 aggregate success.
Athletic Bilbao also knocked out Real Betis and Levante eliminated Rayo Vallecano.
Champions Barcelona take on Getafe on Thursday following a 4-0 victory in the first leg.
(Reporting by Tim Hanlon, editing by Ed Osmond)
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