United went ahead before halftime as Christian Eriksen scored his fourth goal of the season.
But Fener fought back four minutes after the restart when Youssef En-Nesyri headed home Allan Saint-Maximin’s cross from close range.
A penalty call for Fener was then denied on the hour mark as Bright Osayi-Samuel went down under a challenge by United midfielder Manuel Ugarte.
Ref Clement Turpin rejected Fener's claims and was supported by VAR. The decision led to a furious reaction from Fener coach Mourinho, who was eventually sent to the stands by Turpin.
Both Rasmus Hojlund and Alejandro Garnacho had chances to win the game for the visitors, but were wasteful at the key moment.
Antony, meanwhile, was forced off injured after landing awkwardly attempting a flick.
United have three points after drawing their opening three games in the competition while unbeaten Fenerbahce have five points.