The Turkish outfit led from start to finish in Szeged to improve to 2-2 in Group D while dropping the hosts to 0-3.
Nicole Powell led the Fenerbahce scoring parade with 16 points while Vaida Sipaviciute had a game-high 21 points in defeat for Szeviép.
The visitors appeared to be cruising to victory after going in front by as many as 15 in the fourth quarter.
Szeviép kept the faith and chipped away, though.
With 2:18 remaining, Sipaviciute scored and was fouled on the shot.
She converted the three-point play to cut Fenerbahce's lead to 70-66.
Neither team, however, scored the rest of the way.
Emöke Füresz missed a three-pointer, Iva Ciglar two lay-ups and Zuzana Horvath a three-pointer as Szeviép narrowly missed out on finishing off a terrific comeback.
Fenerbahce were looking for their first win since beginning the EuroLeague Women with a victory over Lotos Gdynia.
Haydar Kemal Ates' team stormed out to an 18-5 lead by the end of the first quarter with Turkey's national team twin towers Nevriye Yilmaz and Nevin Nevlin wreaking havoc on the offensive end.
Yilmaz had a three-point play before Penny Taylor passed to Powell and the American buried a jump shot for an early 5-0 lead.
Olena Zherzherunova opened Szeviép's account with a three-ball only for Nevlin to reply with consecutive baskets for a 9-3 Fenerbahce advantage.
Yilmaz hit another jumper to extend the lead to 11.
The home team struggled when they had the ball and only scored their second basket through Füresz with 19 seconds remaining in the first quarter, but Taylor capped a strong Fenerbahce opening 10 minutes with a shot from behind the arc to give the visitors an 18-5 lead.
When play resumed, the hosts picked up the intensity and went on a 10-2 run, with Edit Laczi's jumper reducing the arrears to 20-15 with 7:23 to go in the first half.
Tammy Sutton Brown then dealt Szeviép two blows with consecutive three-point plays for Fenerbahce to put them back on top at 26-15.
The hosts managed to claw the deficit back to 26-20 but went to half-time trailing 40-27 following a pair of Esmeral Tuncluer free-throws.
Szeviép continued to battle away and when Hungary international Horvath drilled a three-pointer, the Fenerbahce advantage was cut to 45-40.
Tuncluer answered with a jump shot and Yilmaz followed with a lay-up as Fener maintained a comfortable lead.
When Tuncluer split two more free-throws at the end of the quarter after a foul by Laczi, the visitors were in command at 61-48.
They had to hold on over the last 10 minutes and did, much to the disappointment of the Hungarian crowd.
Haydar Kernal Ates, Fenerbahce coach:
"It's always hard to win an away game at the EuroLeague. I'm not statisfied with the way we played, we need to do better in the next games."
Tamás Bencze, Szeviép coach:
"We're sorry about the first quarter, when we show too much respect to our opponent, and couldn't play our game at all."