The match on Panathinaikos's court at Nikos Galis Olympic Indoor Hall saw a resilient Fenerbahce side leading 16-15 at the end of the first quarter.
Fourth-seeded Panathinaikos, having never lost at home against Fenerbahce, took control in the second quarter to lead 42-28 at halftime.
The second half belonged to fifth-seeded Fenerbahce, which dominated its opponent 20-8 in the third quarter behind efficient offense from guard Bogdan Bodganovic and a strong defensive presence from center Ekpe Udoh.
In the final quarter, Fenerbahce held Panathinaikos to only 8 points.
Bogdanovic finished with 23 points, 5 rebounds and 4 assists and was designated Most Valuable Player of the match.
Nikola Kalinic added 16 points and Bobby Dixon chipped in 13 for the winners.
"We played excellent on defense, everyone contributed on offense, scoring some easy baskets in transition," Bogdanovic said after the match.
The second match will be played in Athens on Thursday at 9 p.m. Turkish time (1800GMT).
In the other match of the evening, CSKA Moscow defeated Spain's Baskonia Vitoria Gasteiz 98-90 to take a 1-0 lead in the series.