Olexandr Zinchenko's early strike was cancelled out by James Wilson's header, his third goal in two games in the competition, but Oleksandr Zubkov hit a late winner to break the visitors' hearts.
The Reds struggled to make much headway against well-organised opposition in the first half but were unlucky to fall behind in the third minute. Skipper Zinchenko's free-kick from the right appeared to deflect in off Liam Grimshaw to hand the hosts a perfect start.
Pierluigi Gollini was forced to make an outstanding save to thwart Mykyta Adamenko and Nicholas Ioannou cleared off the line from the resulting corner as Nicky Butt's side fought to keep the deficit to a single goal.
Although James Wilson was just unable to get his head to an Evans centre, there were few scares for Donetsk and Denys Arendaruk spurned a chance to add a second before the break. But United improved markedly in the second half and started to show their true selves.
Joe Rothwell skied over and Ben Pearson's cracking effort from distance flew just wide. James Weir was next to try his luck but failed to provide a finish after collecting Pearson's neat pass. After Andreas Pereira had a free-kick beaten out by Bogdan Sarnavskiy, Butt replaced defender Evans with striker Ashley Fletcher.