Goals in the second-half by Giannis Maniatis and substitute Kostas Mitroglu English champions Manchester City and rivals Arsenal both slumped to defeats in their final Champions League group matches leaving City without even the consolation of a Europa League spot and the Gunners failing to top their group and facing a tough Last 16 draw.
City bowed out with a passionless performance in a 1-0 defeat away to a largely second string Borussia Dortmund side, who topped the group, while City finished bottom with three points, the lowest amount recorded by an English side in the history of the group phase.
Ajax took the Europa League place despite losing 4-1 away at Real Madrid.
Arsenal let slip a 1-0 lead to lose 2-1 away at Olympiakos, who secured a deserved Europa League place while German outfit Schalke 04's 1-1 draw away at Montpellier ensured they edged out Arsenal for top spot.
That will only increase the pressure on the English side's previously untouchable manager Arsene Wenger with their domestic title hopes already all but over as they trail leaders Manchester United by 15 points.
While Wenger may be starting to feel the heat, the pressure eased a bit on Carlo Ancelotti, manager of big spending Paris Saint Germain and a side that Wenger has been linked with, as the French outfit beat Porto 2-1 to leapfrog them and top their group.
"I am sure that now things will quickly get better in the league," said Ancelotti.
"There is no problem in the PSG dressing room, when you have to build a team with a lot of new players, it takes time, that's only natural."
There was to be some consolation for the hard-pressed fans of that group's bottom side Dinamo Zagreb as Ivan Krstanovic gained them their only point of the group phase with a penalty five minutes into time for a 1-1 draw with Dynamo Kiev added on in a game played in heavy snow in Zagreb.
In the other group the only outstanding issue was settled as Russian side Zenit St Petersburg withstood enormous pressure from already qualified seven-time champions AC Milan to win 1-0 away in Italy and take third place and a Europa League spot at the expense of Anderlecht.
Manchester City were undone by Julian Schieber, who crowned his first Champions League appearance of the season to score Dortmund's winner when he slotted home from a brilliant low cross by Polish star Jakub Blaszczykowski in the 57th minute.
It left City's outstanding goalkeeper Joe Hart seething.
"I would love to be able to answer why we have been so poor in Europe this season but I can't. All I know is that we have to address it," said Hart.
"We have let ourselves down. We should be competing for a place in the last eight not going out like this."
Real Madrid eased to victory over Ajax with Brazilian star Kaka's superb effort making him the most prolific all-time Brazilian scorer in the Champions League with 28 edging ahead of fellow 2002 World Cup winner Rivaldo.
Real coach Jose Mourinho said that he wasn't disappointed with their overall group campaign.
"Well we emerged from the Group of Death and there will be plenty of teams who topped their groups who will not wish to be drawn against us," said Mourinho.
Arsenal's hopes of keeping their 1-0 lead to the end - thanks to Tomas Rosicky's first goal since March - never looked likely as they had failed to keep a clean sheet on their five previous visits to Greece.
Sure enough goals in the second-half by Giannis Maniatis and substitute Kostas Mitroglu ensured the hosts gained a second successive home win over their opponents, having beaten them 3-1 at the same stage last season.
Wenger took the defeat on the chin, something he is having to get used to of late.
"We had a good first half but dropped physically in the second. Many of the players are not used to that level.
"We have to wait for the draw but ideally you would want to finish first.