It did not take long for the Verdiblancos to break the deadlock, although the opportunity to do so was gifted to them by Costa, who was caught in possession just outside his own area by Sevilla.
The 29-year-old midfielder then weighted a pass through to Molina, who swept confidently past Diego Alves.
Intricate build-up play from Juanfran and Molina unlocked the Valencia defence for Sevilla to burst through and double the advantage in the 22nd minute.
Another wonderfully crafted goal saw the hosts out of sight before half time, as this time a reverse pass from Alvaro Vadillo sent Sevilla darting late into the box to notch his second of the game.
Betis took their foot off the pedal in the second half, allowing the visitors to enjoy the lion's share of possession.
Costa's powerful header gave Dukic's men a lifeline with 23 minutes remaining, but it proved to be nothing more than a consolation goal.
The 3-1 victory marks the first win of the new season for Betis, lifting them to 11th in the table. Meanwhile, Valencia are down in 15th with three points from their four games so far.