It was the Dons' heaviest European loss in 20 years.
Panathinaikos applied the early pressure winning a series of corners that Aberdeen failed to clear but it was the visitors who had the first real chance in front of goal when Chris Clark struck from the edge of the area after picking up a Derek Young cross.
However, his shot went well over the crossbar and the Greeks took the lead on 11 minutes from a free-kick.
The Dons were penalised when Andrew Considine committed a foul level with the edge of the area and Jamie Langfield stood no chance as Ioannis Goumas whipped the ball in at the near post.
Panathinaikos could have made it 2-0 a couple of minutes later from another free-kick but Marcelo Mattos put his effort narrowly over the crossbar.
And Langfield was called into action again midway through the first half when he pushed a Dame N'Doye shot wide of the target to keep his side in the tie.
Aberdeen felt they were being hard done by when Scott Severin was booked on the half-hour mark by Austrian referee Thomas Einwaller for a challenge on the ever-dangerous N'Doye.
But the visitors were awarded a free-kick as the first half drew to a close although Barry Nicholson could not find the target with his effort.
Panathinaikos continued to look the more likely side to score in the second half. Michael Hart did well to block a N'Doye effort early on while ex-Hearts man Takis Fyssas, making his first start of the season, hoofed an effort high over the target.
But the home side did stretch their lead midway through the second half when substitute Dimitrios Papodopoulos made a well-timed run into the box and picked his spot firing through Langfield's legs to make it 2-0.
With Aberdeen now on the back foot, the Greeks put the game out of reach on 77 minutes from another free-kick.
Andreas Ivanschitz's initial curling effort was punched away by Langfield, but Dimitrios Salpigidis pounced on the rebound with the goalkeeper stranded.