Needing a win to avoid an embarrassing early exit, Real dominated throughout and came out on top thanks to 16 points by forward Jeffery Taylor and 14 from guard Sergio Rodriguez.
"We made a poor start in a fiercely competitive group but hit top gear when we needed to and I expect better results in the Top 16," Real coach Pablo Laso told Euroleague television.
Real squeezed through after winning their last three games to finish the preliminary group stage with a 5-5 record.
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Five-times winners Maccabi failed to reach the second pool stage, featuring two groups of eight, after falling short in their bid to edge Darusaffaka on a better head-to-head score in their section.
Both teams finished with a 4-6 record but the Turkish side advanced in their maiden Euroleague season on the back of a shock 84-73 win in the reverse fixture.
Maccabi had the upper hand in the opening three quarters but fell behind 62-52 in the final period and ran out of time to turn circumstances in their favour with a late rally.
"We worked hard for this and deserve to be among Europe's best teams," said Darusaffaka forward Milko Bjelica.
Red Star Belgrade and Bayern Munich clash on Friday in a dogfight for the last remaining Top 16 berth, with tickets sold out for the mouth-watering fixture in the Serbian capital's Pionir Arena.
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