With questions being raised over David Moyes' credentials as Sir Alex Ferguson's successor following a miserable start to the Premier League season, the Scot will have taken heart from a much-improved performance following the defeat by West Brom, despite the absence of shin-injury victim Wayne Rooney.
After a cagey opening, the first chance of the game came with eight minutes on the clock when Tom Cleverley's scooped ball forward found Robin van Persie on the edge of the box. The striker turned swiftly away from his marker only to curl the resulting shot over Andriy Pyatov's crossbar.
The home side responded, pushing United back, and Cleverley was extremely fortunate to avoid conceding a penalty when his badly-timed sliding challenge clipped the ankle of Alex Teixeira, sending him tumbling to the turf, but perfectly-positioned referee Pavel Kralovec was unmoved.
There was a stroke of good fortune about United's opener in the 18th minute which began with an incorrectly awarded throw-in and ended when Welbeck pounced from close range as Yaroslav Rakitskiy failed to cut out Marouane Fellaini's cross at the near post following an excellent turn and surge from the midfielder.
There were few alarms for David de Gea as the half progressed, although he had to be alert to smother a shot from range from Teixeira seven minutes before the interval, before Chris Smalling failed to deal with a routine through-ball, only for the keeper to gather ahead of Luiz Adriano.
Rakitskiy attempted to make amends for his earlier blunder at the start of the second half, firing in a powerful free-kick which initially had De Gea scrambling before the keeper watched it flash past the post, with the United No.1 called into action again as he bravely gathered Vyacheslav Shevchuk's low cross under pressure from Fernando.
Fellaini went into the book just after the hour after he and Antonio Valencia challenged Taison, with the Belgium international midfielder's reaction suggesting it was his team-mate who had brought down the Shakhtar man.
Nemanja Vidic was cautioned soon after as he made absolutely sure Douglas Costa would not get away from him on the edge of the box by taking a handful of the forward's shirt and hauling him to the turf, with Darijo Srna firing the resulting free-kick into the wall.
Giggs came on for Fellaini to make a piece of Champions League history in the 66th minute, surpassing Raul as the record appearance maker in the competition with his 145th outing among Europe's elite.
But United's hopes of an impressive three-point haul, on a ground where no English side has won in the Champions League, were dashed with 14 minutes remaining as Rakitsky foraged forward and fired over a low cross which Vidic could only deflect out into the path of Taison, who venomously lashed home beyond a helpless De Gea.
The United keeper produced an excellent near-post stop with time running out and his side under increasing pressure as he reacted sharply to turn away Taison's cross after it was deflected goalwards by Smalling.