The Spaniards were in control from the start of the match, but fell 1-0 behind after the Greeks scored in the 59th minute through a thunderous drive from 25 metres by Giorgos Karagounis.
Villarreal levelled eight minutes later when Giuseppe Rossi calmly converted a penalty, awarded after substitute Robert Pires was brought down in the box by Jakub Wawrzyniak.
But the Spaniards rued their missed chances.
"It is incredible that we failed to win this match," midfielder Santi Cazorla said. "We were the better team throughout the game, far more dangerous."
"Now we are forced to score a goal away and qualify for the quarter-finals," he said,
Panathinaikos coach Henk Ten Cate fielded an ambitious 3-6-1 formation in an effort to contain Villarreal's frantic pace.
The hosts surged forward from the start, coming close through Joseba Llorente who did well to chase down a ball on the right after four minutes but failed to beat Panathinaikos keeper Mario Galinovic.
Rossi was ready to score in the 21st only to see Loukas Vintra tackle him at the last moment.
The hosts had another chance through Sebastian Eguren when Galinovic failed to clear a cross but his header from three metres flew wide while Ariel Ibagaza's left-footer in the 32nd minute scraped the post.
The keeper also did well to save a Rossi effort from point-blank range as the Spaniards desperately tried to break the deadlock before the end of the half.
THUNDERBOLT
Villarreal went on the attack in the second half, coming close again through Cazorla but it was the visitors who grabbed the lead when Karagounis picked up the ball some 25 metres out and gave Villarreal keeper Diego Lopez, who got his fingertips to it, little chance.
Vangelis Mantzios could have scored a second for the Greeks minutes later but Lopez was well positioned to block his close-range effort.
Villarreal coach Manuel Pellegrini brought on experienced Nihat Kahveci and Robert Pires three minutes after the goal and was rewarded in the 67th when the former French international was brought down in the box.
Rossi sent Galinovic the wrong way to level.
"We got what we wanted," Dutchman Ten Cate said. "We got an away goal, we didn't lose against a very good team that played at times fantastic football.
"Villareal are a better team than us," Ten Cate said.
The hosts piled on the pressure in search of a winner but their efforts through Rossi, Kahveci and Bruno were saved by Galinovic.
