Dos Santos himself came close shortly afterwards when he drove into the box before drawing a save from Andres Fernandez, who then recovered in time to deny Moises Gomez on the rebound.
The first goal of the game came less than three minutes after the break when Jeremy Perbet collected Jaume Costa's pass before sliding the ball under the keeper to give his side the lead.
Things got even better for the hosts just eight minutes later, as Manuel Trigueros found space on the edge of the box to fire a low effort into the bottom corner and double his team's advantage.
Dos Santos almost got his name on the scoresheet with 15 minutes left when his curling effort towards the top corner drifted narrowly the wrong side of the post.
They clinched the victory moments later, however, as Perbet latched onto Gomez's pass before beating Fernandez for his second goal of the game.
Osasuna did get themselves on the board late on. Oriol Riera converted Javier Acuna's cross just four minutes after coming on as a substitute, but it was too little too late for the visitors as Villarreal saw out the victory.