Roberto Trashorras squeezed his 91st-minute spot kick under Diego Lopez for a 1-0 victory on the night, after home defender Angel was adjudged to have tripped Pedro Botelho as he raced into the area.
Celta, who drew 1-1 at home in the first leg, moved into next week's quarter-finals where they will meet Atletico Madrid or second division Recreativo Huelva.
"I was a long way from the penalty and I don't know if it was inside or outside the area, or whether he touched the player or not, but I'm not interested now because we are out," Villarreal coach Ernesto Valverde told reporters.
"I don't want to look for excuses. You can't call their qualification for the quarters unfair."
Holders Barcelona need to overturn a 2-1 deficit when they visit Sevilla on Wednesday in the last-16 round's highest- profile tie.
