The much-changed La Liga side had suffered just one defeat in their last 10 matches in all competitions since October 1, and they went ahead after just 13 minutes when Kike Sola fired home from 15 yards.
Aymeric Laporte doubled the lead just six minutes later when he headed home Ander Iturraspe's corner from close range, and though both sides had plenty of chances, that early double was enough for the Spanish Super Cup winners to take a two-goal lead back to San Mames.
However, with the lower-seeded clubs enjoying home advantage in the first leg, there were a few surprises in Thursday's action as three La Liga sides were beaten.
Segunda Division strugglers Huesca, playing in the last 32 for the first time ever, triumphed over Villarreal in a five-goal thriller thanks to Darwin Machis's winner 15 minutes from time, despite Christian Fernandez's injury-time red card.
Luis Fernandez had given the hosts the lead after just three minutes, before three goals in eight minutes early in the second half saw them twice pegged back, Nahuel Leiva and Cedric Bakambu scoring for Villarreal either side of Fran Merida's penalty.
La Liga bottom side Malaga were also beaten on the road, as Junior Lago's goal secured victory for second-tier side Mirandes, despite Roque Santa Cruz cancelling out Alejandro Garcia Pena's first-half opener shortly after the break.
Meanwhile, Ponferradina, also of the Segunda Division, stunned top-half La Liga contenders Eibar with a 3-0 victory to put themselves in control of their tie thanks to goals from Luka Djordjevic, William Jebor and Dmytro Khomchenovsky.
In the night's only all-top-flight clash, Levante struck late on through substitute Roger to hold Espanyol to a 1-1 draw at home after Mamadou Sylla Diallo had given the visitors the lead midway through the second half.
Copa del Rey results in full (half-time scores in brackets): Huesca 3-2 Villarreal (1-0), Levante 1-1 Espanyol (0-0), Linense 0-2 Athletic Bilbao (0-2), Mirandes 2-1 Malaga (1-0), Ponferradina 3-0 Eibar (2-0)