Nicolas Pépé gave the hosts an early lead before Luismi Cruz levelled from an Aubameyang assist, and an Adama Traoré own goal restored Villarreal's advantage before Zakaria Eddahchouri equalised again, also assisted by Aubameyang, ahead of the Gabon striker's own late winner.
The result leaves Íñigo Pérez's side still winless from four league games ahead of their Champions League trip to Borussia Dortmund, while promoted Deportivo remain unbeaten.
Pérez was reluctant to hide behind refereeing or individual errors when assessing the defeat.
"I agree. But beyond the errors that can happen in football, we have to look at and correct other things, because even without making those mistakes and finding a win, the feeling would have been similar," he said. "I think we've been missing something that, as a coach, I'm looking for and maybe haven't found."
The Villarreal boss was clear that the club's poor start needs urgent attention before Wednesday's Champions League opener.
"The player is capable of regaining confidence within two days, knowing they've done it many times before, that they've had success, that they create chances, that they can win in Dortmund," he said.
"That doesn't worry me and I know they'll be ready, but this start to the season gives us some symptoms, some alarm bells, that we need to address urgently so they don't repeat themselves."
Pérez did find one positive to take from the afternoon in the performance of a new arrival.
"One of the positive aspects has been the emergence of Nathan. I think he had a very important game. With little time on the pitch he's shown energy, shown quality, and I think that's something to hold on to," he said, referring to summer signing Nathan Saliba.
Antonio Hidalgo, meanwhile, was thrilled to see his newly promoted Deportivo side claim a prestigious scalp.
"This is a very long road, there's less left for survival, but there are still many points needed for the club's objective, which is staying up," the Deportivo boss said. "We've got a brutal victory against a great team."
Hidalgo praised his players' courage after twice falling behind.
"In the end, the rivals in the top flight demand a lot from you, you have to be brave, and we have bravery and a cutting edge, and that's a great virtue," he said, before turning to his star forward.
"You have to see how he trains and the smile he spreads to everyone. I'm lucky to have hungry, high-quality players — we have to keep working hard and fight for three points against Getafe next."