The Welshman, who joined Madrid from Tottenham for a reported world record fee of around £80 million ($124 million, 94 million euros) earlier this month, turned home Dani Carvajal's cross five minutes from half-time to send Madrid in level at the break after Cani had given Villarreal a deserved lead.
Bale was then withdrawn to make way for Angel Di Maria on the hour mark, but the visitors were in front just two minutes later when Cristiano Ronaldo's effort ricocheted past Sergio Asenjo in the Villarreal goal.
However, Giovani Dos Santos, another former Tottenham player, swept home after Diego Lopez had denied Cani a second 21 minutes from time to preserve Villarreal's unbeaten record on their return to the top flight as Real fell two points behind Atletico Madrid and Barcelona at the top of the table.