Forward Martin Vingaard put the Danes ahead in the 26th minute from the right side of the area, lashing in a half-volley fro 20 metres with their first shot on target.
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Panathinaikos then began to look more dangerous on a frosty night but five minutes after the break Nikos Spyropoulos brought down Copenhagen's Jesper Gronkjaer as he broke clear on goal.
There was a sigh of relief from the home fans as the winger got up to make it 2-0 from the penalty spot.
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The home side, roared on by an enthusiastic 36,000 crowd, should have scored again when Dame N'Doye went round goalkeeper Alexandros Tzorvas but sent his shot against the post.
They sealed victory in the 73rd when Panathinaikos captain Djibril Cisse nodded an own goal off a Vingaard corner.
Panathinaikos, who finished bottom of the group, grabbed a late consolation goal through Cedric Kante's header in stoppage time, the Greek side's 300th goal in 245 European games.
Copenhagen coach Stale Solbakken said he was "happy and relieved" after his side had wasted opportunities to pick up points near the end of the group stage.
Copenhagen finished with 10 points behind unbeaten Barcelona on 14 and are only the second Nordic team after Norwegian club Rosenborg to make it so far in the competition's current format.
MATCH POINT
"When we started... we could only cause a surprise but towards the end of the group stage we twice had 'match point' and that increased the pressure on us so I'm happy and relieved that we succeeded," said Copenhagen coach Stale Solbakken.
His side's previous 'match points' came in a 1-1 draw at home to Barcelona and a 1-0 defeat at Rubin Kazan.
"I dare say that we have maybe one of the best defences in the entire Champions League," added Solbakken. "At the same time, we play a good and solid basic game, and I think we were the second best team in this group.
Looking ahead to the knockout phase, he told reporters: "We need to avoid Real Madrid and the three big British teams to go beyond the next stage.
"But if I were to point out some opponents who we would have the best chance against I would say Schalke, Tottenham or Shakhtar Donetsk, if those teams are among the opponents we could draw."
Panathinaikos coach Jesualdo Ferreira, who took charge in the middle of the campaign, turned his attention back to the domestic league.
"We didn't have enough time to change things," he said.
"Now we're number one in our country and should focus on staying on top in Greece."
