The Magpies produced an impressive performance and it looked like they would emerge victorious until Christian Baroni struck for the Turkish side just before the final whistle.
Fenerbahce carved out several chances to open the scoring, with former Liverpool man Dirk Kuyt among those going close with a volley that hit the side-netting.
Newcastle still posed an attacking threat of their own, with the fit-again Sylvain Marveaux catching the eye before being substituted.
The deadlock was finally broken in the 76th minute when Abeid collected the ball from Gael Bigirimana, held off a defender and slotted home.
Shola Ameobi and Papiss Cisse had chances to extend Newcastle's lead before Baroni's first-time effort beat Rob Elliot and flew into the bottom corner.