The visitors got off to a flying start, building a 24-15 lead after the opening quarter.
But Anadolu Efes fought back into the match thereafter, with Ersan Ilyasova scoring 21 points, before finally falling one point short at the end of a nailbiting match.
In the day's other matches, Mirza Teletovic starred for Caja Laboral as they beat Bennet Cantu 81-69 at the Iradier Arena to record their second win of the season.
There were also second wins of the Euroleague for Panathinaikos and CSKA Moscow.
KK Zagreb star Sean May saw his 23 points scored in vain as his side lost 81-62 against Panathinaikos in Zagreb.
And Nenad Krstic was in blistering form, sinking nine-out-of-nine attempts on the basket as CSKA smashed Brose Baskets 94-74 in Moscow.