United boss David Moyes made nine changes to his side for the trip to Ukraine, with Wayne Rooney missing out due to a shin injury.
Robin van Persie was a notable inclusion for the visitors and started brightly, smashing over from close range after controlling a floated pass from Tom Cleverley.
Shakhtar carried a threat with pace down both flanks and had a penalty appeal turned away early on, with Cleverley breathing a sigh of relief after clipping Alex Teixeira, although the forward did fall theatrically.
A clinical counter then saw United take the lead after 18 minutes as Welbeck broke the deadlock from close range, making the most of Yaroslav Rakitskiy's miscue from a Marouane Fellaini cross.
Shakhtar looked livelier in the last five minutes but United largely controlled the first period, holding on to their narrow advantage heading into the break.