Danny Welbeck opened the scoring for the Red Devils, but a second-half equalizer denied Moyes his second Champions League victory.
Ryan Giggs made history by breaking the UEFA Champions League appearance record after coming on to replace Marouane Fellaini.
But Danny Welbeck was the initial hero as he gave United the lead in the 18th minute before Taison equalized with 14 minutes remaining — but the game was more notable for Giggs' landmark appearance.
The Welshman replaced Fellaini in the 66th minute to take him to 145 appearances in the competition, one more than Raul.
Shakhtar coach Mircea Lucescu made two changes from Saturday's 2-2 draw with Metalurh Donetsk, as Olexandr Kucher and Luiz Adriano came in for Sergey Krivtsov and Facundo Ferreyra.
Lucescu had stated before the game that he felt David Moyes was struggling because he had not yet found his best starting XI and, again, the United manager made wholesale changes from the home defeat against West Brom at the weekend. Only goalkeeper David de Gea and Michael Carrick kept their place from Saturday, with Wayne Rooney missing out with a shin injury picked up in training Tuesday.
After a quiet opening, Robin van Persie had the first sight of goal in the eighth minute, bringing down Tom Cleverley's pass before firing high and wide from 20 yards.
There was little further goalmouth action until the visitors took the lead in the 18th minute when Fellaini drove to the byline before cutting the ball back in search of Welbeck. Yaroslav Rakitskiy was in position to intercept a pass but the defender slipped as he stuck out a leg, allowing the ball to run for Welbeck to sweep beyond Andriy Pyatov from 6 yards.
The home side was struggling to create anything going forward and did not manage an effort on target until the 39th minute when Alex Teixeira's weak shot from the edge of the box was straight at de Gea.
The second half followed a similar pattern to that of the first, Rakitskiy's wayward free kick from 35 yards in the 50th minute representing Shakhtar's only notable effort of the opening exchanges.
Van Persie had a rare sight of goal in the 71st minute, turning on the ball inside the area before seeing his low effort blocked by the covering Rakitskiy.
But five minutes later, the hosts equalized out of nothing. Rakitskiy broke down the left and although Nemanja Vidic intercepted his cutback, the ball fell perfectly for Taison to smash it beyond De Gea from 10 yards.
And Taison then came close to a winner five minutes from time as his deflected drive was tipped over brilliantly by the alert De Gea at his near post.
The result sees both teams move to four points from two games in Group A, although Moyes may be disappointed that his side could not see out a game in which it held the lead.