Fernando Vazquez took over during the week with the tough task of steering the Galician side to safety, and he got a first-hand look of how much work there is to do as Deportivo quickly fell behind on the road to Ivan Rakitic's fifth-minute goal.
Gary Medel got another in the 27th minute and although Ivan Riki soon replied for Depor, any hope of a comeback was quickly snuffed out by Medel's second.
Depor showed plenty of heart as they sought a way back into the game late on, but Antonio Beto made a string of saves to keep them out.
Sevilla took an early lead as Rakitic got up to head home Andujar Coke's cross, and it might have been 2-0 moments later when Daniel Aranzubia made an excellent save to deny Jose Antonio Reyes.
The hosts were relentless with Aranzubia again saving well from Geoffrey Kondogbia before Federico Fazio headed wide.
A second goal was coming and it duly arrived in the 27th minute when Jesus Navas slipped the ball through for Medel to rifle home.
Depor responded well as Riki turned home Abel Aguilar's pass from close range five minutes later but it was not to be the start of a fightback.
Sevilla remained on the offensive with Medel going close with another shot before scoring the third goal five minutes before half-time, slotting home from close range from Alberto Moreno's cross.
Aranzubia held on to Emir Spahic's header as they avoided any more damage before half-time.
Evaldo Santos was presented with a decent opportunity for Depor early in the second half but failed to get any power on his header, but it was a sign of things to come with Depor enjoying the best of the second-half chances without capitalising on them.
Beto made a smart save to turn Luis Pizzi's close-range effort around the post before his 77th-minute header brought an even better block out of the Sevilla goalkeeper as he pushed the ball away from danger.
The goalkeeper then had to make another excellent block to keep out Nelson Oliveira's header from a corner.
At the other end, Aranzubia was alert to deny Rakitic in the closing minutes but the victory was already secure for Sevilla.