Lionel Messi opened the scoring with a fine goal after just three minutes and Andres Iniesta added an even better second later in the half as the Spanish champions quickly took control at El Sardinero.
But the Spain striker struck again after 62 minutes with a pinpoint header to score his first goal in Barca colours and ensure an ideal start to the league campaign for the Catalans.
Villa remarked afterwards: "It is an honour to play in this team and with players like Messi, the best player in the world. Whoever does not fit in with him, doesn't deserve to be in this team.
"I do not manage the squad but we have shown today and in the Super Cup that we are able to do well with the squad we have."