A Muniain goal in each half and strikes from Markel Susaeta and Aritz Aduriz at a rain-lashed San Mames set up a meeting with Real Betis in the next round.
"Although the result suggests it was an easy match it required a huge amount of effort," Aduriz told Canal Plus television.
"The important thing now is to rest and prepare for Sunday's La Liga match at home to Rayo Vallecano so that the squad could sign off for the winter break with another win," said Aduriz.
Athletic, who have won the second-most Copa del Rey trophies (23) after record winners Barcelona (26), were joined in the last 16 by Valencia, Rayo and Espanyol.
Valencia scraped past Gimnastic when they beat the third tier side, who held them to a 0-0 draw in the first leg, 1-0 at the Mestalla stadium.
Still without a coach after Monday's sacking of Miroslav Djukic, Valencia will play holders Atletico Madrid for a place in the quarter-finals.
Rayo and Real Valladolid were also goalless in their first leg before the former won the return in Madrid 3-1. Espanyol beat second division Real Jaen 2-0 for a 4-2 aggregate triumph.
The remaining last-16 ties are Barca v Getafe, Real Madrid v Osasuna, Real Sociedad v Villarreal, Rayo v Levante, Espanyol v Alcorcon and Almeria v Racing Santander.
Second division Alcorcon, who knocked out Granada, and third tier Santander, who eliminated Sevilla, are the only non-La Liga sides left in the competition.
Barca look to have an easier path to the final, with Real Madrid and Atletico on the other side of the draw and on course to meet in the last four.
The last-16 ties and the quarter-finals are in January, the semi-finals in February and the final is at a yet-to-be-determined venue on April 19.