Getafe also took control of their quarter-final with a 2-1 victory at Real Mallorca.
The other two first legs will be played on Thursday when Atletico Madrid host second division Celta Vigo and Osasuna visit Racing Santander.
The 2007 winners Sevilla, who knocked out holders Barcelona last week, took a 26th-minute lead when Alvaro Negredo raced on to a Jesus Navas chip over the top to head the opener.
Juan Rodriguez headed against the post for Depor but it was Renato who doubled the visitors' lead, nodding in after good work from Diego Perotti in the 67th.
Two minutes later, Negredo burst down the right and pulled back for Navas to stroke in the third.
"We needed a game like this and I hope it will be the start of a new run of form," Negredo told Canal Plus television. Sevilla had lost their previous three games.
At the Ono Estadi, second-half goals from Manu del Moral and new Venezuelan signing Nicolas Fedor, known as 'Miku', gave the 2007 and 2008 finalists Getafe a comfortable lead over Mallorca.
But the islanders staged a late rally, pulling one back through Uruguayan striker Gonzalo Castro in stoppage-time, though it was not enough to maintain their 100 percent home record this season.
Mallorca, fourth in La Liga, are the highest-placed team left in the competition with Barca, Real Madrid and Valencia all knocked out.
