One of the poorest teams in the Spanish top flight and relying on one of the oldest squads, Levante are a surprising fourth in La Liga and briefly topped the standings for the first time in their history in October.
Wednesday's resounding cup win set up a last-eight clash with their more illustrious city neighbours Valencia.
The 2008 winners lost 2-1 at Sevilla, the 2010 champions, in their last-16 second leg at the Sanchez Pizjuan but went through on the away goals rule after Valencia won last week's first leg at the Mestalla 1-0.
"Our priority is the league but we are going to try to compete in the cup too," Levante coach Juan Ignacio Martinez in his first top-flight coaching job, told a news conference.
"If we get through the next round we'll be very pleased and hopefully we can go far, although it's going to be very tough."
Ruben Suarez saved the best until last when he whipped a stunning free-kick into the top corner from 30 metres for the home side's fourth 24 minutes from time.
Cordoba battled back to 3-2 with goals from Alberto Aguilar and Pepe Diaz but Vazquez ended the Andalusian club's hopes when he lifted the ball over the Cordoba keeper and into the net to send the 2006 winners through.
Espanyol will play third-tier Mirandes in the last eight after the Burgos-based club pulled off an upset at Racing Santander on Tuesday when they drew 1-1 at the Sardinero after winning last week's first leg 2-0.
Jose Mourinho's side will play their great rivals Barcelona, the team they beat 1-0 in last year's final, in the quarters as long as Barca can defend a 4-0 lead at Osasuna on Thursday.
Bilbao have home advantage for Thursday's ret
