Sevilla's chances on a badly cut-up pitch at the Mestalla were dealt a blow shortly before the break when Mali striker Frederic Kanoute was shown a second yellow card for a dangerous tackle on Valencia captain Carlos Marchena.
Two minutes after the restart, Mata drove the ball high past Sevilla goalkeeper Andres Palop from the edge of the box and Hernandez made sure of victory with 10 minutes left when a first-time shot off his shin flew into the net.
"We passed the exam today but we need to keep studying," Valencia coach Unai Emery told a news conference.
Atletico Madrid's campaign got off to a poor start earlier on Sunday when they were thrashed 3-0 at Malaga.
Villarreal squandered a 1-0 lead at Osasuna to draw 1-1, while promoted Xerez lost 2-0 at Real Mallorca and had two players sent off in their first match in Spain's top flight.
The scale and manner of the defeat puts unwelcome pressure on Atletico coach Abel Resino, who was sent to the stands midway through the first half for protesting the referee's decision to caution Argentina forward Sergio Aguero.
"That was Atletico Madrid's worst performance since we came back from vacation," Resino told a news conference.
"We need to keep working and correct the mistakes because this league has just started," he added.
"As far as my sending off goes, I just told the referee that Aguero didn't deserve the card."
Barca's opening match of the season against Sporting Gijon has been put back to Monday due to the Spanish and European champions' European Super Cup match against Shakhtar Donetsk on Friday, which they won 1-0.
Real Valladolid managed to hang on for a 0-0 draw at Almeria in Sunday's late match despite having Brazilian defender Nivaldo sent off in the 13th minute.
