Macedonia, who were also fancied to make the semi-finals of the 12-team tournament in Venezuela, were sent packing after an 86-76 defeat by the Dominican Republic while Russia and Lithuania rolled into the last four.
The Russians, 80-65 winners over Angola, will play Nigeria for a berth in the final, while Lithuania, who eked out a 76-72 success against Puerto Rico, will take on the Dominican Republic later on Saturday.
The top three teams will qualify for the Olympics and join the United States, Great Britain, Spain, France, Argentina, Tunisia, Australia, Brazil and China.
The Greeks seemed to be on course to qualify when they opened up a 13-point lead midway through the third quarter but Nigeria's bench player Ade Dagenduro dashed their hopes by scoring the last seven points for his team in a frantic finish.
The defeat will have come as a huge blow to Greece, who last won the European Championship in 2005 and whose champions Olympiakos Piraeus are also the title holders in the Euroleague, which is the continent's elite club competition.
American-born point guard Lester McCalebb buried 25 points for Macedonia, who finished fourth in last year's Euros in Lithuania, but his valiant solo effort was not enough to contain a rampant second-half performance by the Dominicans.
Fernando Garcia led the winners with 28 points while Atlanta Hawks forward Al Horford recovered from a poor first half to finish the game with 14 points and 15 rebounds.