The Spain striker headed the opener in the 36th minute, and after Ivan Rakitic had doubled the lead just before halftime, he bagged his second from the penalty spot. Manu del Moral completed a 4-0 aggregate win in stoppage time.
Zaragoza had defender Jose Fernandez sent off with a straight red card after only 26 minutes, and midfielder Fran Gonzalez followed him down the tunnel in the 66th, after giving away the penalty for Sevilla's third.
The 2010 Cup winners will meet either Atletico Madrid or city rivals Real Betis in the last four.
Atletico, who lost against Sevilla in the final three years ago, hold a 2-0 advantage ahead of Thursday's trip to play Betis.
The game started badly for Zaragoza at a half-empty Sanchez Pizjuan with fullback Fernandez being sent off after flying into a challenge and catching Jose Antonio Reyes in the chest with his knee.
A few minutes later, they also lost striker Javi Alamo to a knee injury, and the hosts took full advantage when Negredo powered in a header from a Rakitic corner.
The Croatia midfielder effectively settled the tie when he sped down the left flank and fired the second high into the net.
Zaragoza's Gonzalez tripped Cicinho in the area to pick up his second yellow card, and Negredo netted from the penalty spot.
Del Moral tapped in the fourth to crown a first win for new Sevilla coach Unai Emery, since he took the helm last week.
Real Madrid survived red cards for Fabio Coentrao and Angel Di Maria, and an injury to Iker Casillas, to draw 1-1 in Valencia, and qualify for the semi-finals. They will face either holders Barcelona or Malaga, who meet at the Rosaleda on Thursday with their tie level at 2-2.