Valencia, who only survived relegation by two points in the last campaign, took the lead on the hour mark thanks to a header from Guinea international defender Mouctar Diakhaby after good work by Portuguese teammate Andre Almeida.
Sevilla levelled nine minutes later when Moroccan striker Youssef En-Nesyri headed in at the far post from a cross by Suso.
However, Valencia stunned the home side in the 88th minute when Javi Guerra hit the winner after Hugo Duro had set-up the opportunity.
Sevilla played the last 10 minutes of the match with 10 men after Frenchman Loic Bade was red-carded for shirt pulling.
Earlier Friday, Rayo Vallecano came away from Almeria, who survived last season by just one point, 2-0.
Both goals came from the penalty spot with Isaac Palazon and Randy Nteka on target in the first half.
Defending champions Barcelona get their campaign underway on Sunday at Getafe.
Real Madrid go to Athletic Bilbao on Saturday while Atletico Madrid host Granada on Monday.
Kylian Mbappe, who is heavily linked with Real Madrid, has refused to sign a contract extension with PSG where his contract ends in 2024.
Josko Gvardiol is one of the most highly rated defenders in the world and enhanced his reputation with impressive performances for Croatia at the World Cup last year.
Brighton & Hove Albion goalkeeper Robert Sanchez is also reportedly close to joining Chelsea while Manchester United are also pursuing Real Madrid’s Aurelien Tchouameni.