He found Mason Greenwood on the left-hand side of the box, and the United forward cut inside before finding the bottom corner with a composed finish to give United the lead.
Fenerbahce responded well to falling behind, and they got their reward in the 29th minute. A corner from the right was met by the head of Osayi-Samuel, who powered his header past Dean Henderson to level the scores.
The hosts pushed for a winner in the second half, but United held firm to secure a valuable point in Istanbul.
Despite the draw, United remain winless in the Europa League this season, with just one point from their opening two games.