The Spain striker, his country's leading all-time scorer, has had to be content with a secondary role this season, but has continued to score when called upon and took his tally to five in four Cup games.
The 31-year-old netted twice in five first-half minutes to ease Barca's passage into the last eight, while a double from Alexis Sanchez after the break to add to the early opener from Thiago Alcantara gave the hosts a 7-0 aggregate victory.
"Villa has played a great game. He has showed us what we all already knew, that he should be an important player for us," Barca assistant coach Jordi Roura said as he stood in for the absent Tito Vilanova.
"He will return to be more important for us, without any doubt."
Barca set up a meeting with Champions League last-16 side Malaga in the next round, and could meet Real Madrid in the semi-finals, if Jose Mourinho's side overcome Valencia.
On the other side of the draw, Atletico Madrid also made the quarters with a drab 0-0 draw at near neighbours Getafe after a 3-0 first-leg win.
Their task was made easier by a red card for Getafe defender Rafa Lopez after 33 minutes and they will next meet Real Betis after the Andalusians scraped through 2-1 on aggregate against second division Las Palmas thanks to a late winner.
Betis had to work hard for their victory, having drawn 1-1 in the Canary Islands, and had to wait until five minutes from time before Ruben Castro pounced on an error to fire the only goal of the second leg.
Las Palmas's strikers Macauley Christanus and Thievy will look back on some glaring misses.
Real Zaragoza take on 2010 winners Sevilla in the remaining quarter-final. The first legs are to be played next week.
Barca had left players such as Gerard Pique, Xavi and Andres Iniesta out of the squad and newly-crowned World Player of the Year Lionel Messi stayed on the bench, but they still dominated against Cordoba.
Thiago burst though the middle to slot home the first after 17 minutes, and Villa scored a superb second soon after.
The former Valencia frontman showed quick feet to skip around a defender in the area, and then from a tight angle dinked the ball over the keeper.
Five minutes later, he fired a snapshot in off the far post.
Chile striker Alexis, one of the players keeping Villa on the sidelines, headed a fourth soon after the restart and grabbed his second and Barca's fifth in the 85th.
The evening ended on a sour note for Villa as he was forced to limp off with what looked like a thigh muscle problem 12 minutes from time, though the club later said it had only been as a precaution.
- Sports & Recreation