The Italians conceded three goals in 13 minutes thanks to a handful of errors which saw them lose out on a group-stage place to Athletic Bilbao
Marek Hamsik handed the Italians the lead just after half-time at the San Mames, but three goals in 13 minutes turned the Champions League playoff on its head.
Aritz Aduriz scored when unmarked from a corner before slotting into an empty net after a calamitous mix-up between Rafael and Raul Albiol, before Ibai Gomez capitalised on a failed offside trap to seal the win.
"Athletic deserved to score in the first half and didn't," Benitez told Sport Mediaset after the match. "We took the lead, but in a few minutes we made some important mistakes and lost control of the game. We made mistakes on all three goals.
"Those goals allowed Athletic to gain confidence and we lost ours. Do I blame anyone? No, mistakes happen - they are not normal, but they happened anyway. We had worked over the week on specifics like corners, but we still made mistakes and clearly we still have work to do.
"We said from the start Athletic were a tough team. We knew it could end up like this, we are all disappointed, but we have to look forward."
Gonzalo Higuain was visibly angered after the draw in the first leg but Benitez would not be drawn on speculation that the striker could now become unsettled after Napoli's failure to reach the group stage.
"I think at this stage we need to concentrate on the game with Genoa on Sunday. We are all disappointed, but cannot think about whether one player is more or less unhappy than the others," he added.