The game exploded into life four minutes into the second half when Argentine playmaker Ibagaza fired past Mario Galinovic to put the Spanish side 1-0 ahead.
Panathinaikos drew level six minutes later with a Vangelis Mantzios header but substitute Joseba Llorente settled the game when he volleyed the winner in the 70th minute.
"I am very happy. It is very important for us to be among the best eight teams in Europe," Villarreal coach Manuel Pellegrini said.
"I think we deserved the result because we also deserved to win the first leg," the Chilean told reporters.
'MORE POSSESSION'
"The first half was very balanced but in the second we were able to press forward more and in that way we gained possession and became more dangerous."
Playing in front of a 70,000 home crowd, Panathinaikos made a confident start and had the better of the chances in the first half, Mantzios producing one fine save from Diego Lopez.
Villarreal served notice of their threat, however, when Ibagaza put the ball in the net only to have a goal ruled out, apparently for a foul in the build-up.
The Argentine made no mistake four minutes into the second half when he curled a shot from the edge of the penalty area past Galinovic.
Mantzios put Panathinaikos back on level terms when he got his head to a Giorgios Karagounis corner from the left.
But, three years after they reached the semi-finals, Villarreal moved into the last eight when Ibagaza laid the ball off for Llorente on the right to shoot past Galinovic.
Panathinaikos coach Henk Ten Cate said: "I can't complain about any of my players. They did everything tonight but it just wasn't enough.
"At this level it's the small details that make the difference and that showed in their second goal when we left three players unmarked. It wasn't very intelligent.
"They killed us with that. We are exiting the tournament with our heads held high," the Dutchman said.
