David Moyes will have felt the sweet surge of relief at the final whistle. This had become the game he dare not lose after consecutive 1-0 defeats to Everton and Newcastle United suffered by his Manchester United side here.

Fifty-one years had passed since the reigning domestic champions last experienced three defeats on the bounce at Old Trafford. In the days leading up to the Cuban missile crisis, Manchester City, Burnley and Blackburn Rovers all left the United cathedral victorious and, with the fierce scrutiny on Moyes, for that record to be ended by Shakhtar Donetsk would send the crowd home with further doubt regarding Sir Alex Ferguson's successor.

The strike that gave United a first victory of December was a scrappy Phil Jones 67th-minute effort, though that hardly matters as thoughts turn to Sunday's trip to Aston Villa and trying to bridge the seven-point deficit to City and a Champions League berth.

What is of more concern is the sluggish form that caused those reverses last week and which continued here until Moyes took a blast at his side during the interval that did wake his charges, though even then they hardly convinced.

The XI the Scot sent out featured no Robin van Persie, who was on the bench, while Nemanja Vidic, Patrice Evra, Chris Smalling, Danny Welbeck and Marouane Fellaini did not make the squad due to injuries. Moyes stressed these were not serious, though Michael Carrick is still missing as he continues his recovery from a long-term achilles problem.

Tonight was about the result and the performance, following the flat displays against Everton and Newcastle. Given the choice, though, Moyes would plump for victory to ease the pressure and allow breathing space as he attempts to plot a course through the hectic festive programme.

There was scant sign of tension from the manager as he strode to his seat or in the opening move put together by his team, though this would prove the falsest of dawns. Adnan Januzaj roved forward, passed to Shinji Kagawa, and he swept the ball over to Ashley Young only for the sequence to break down when Rafael da Silva mis-hit across.

Jones, again partnering Ryan Giggs in central midfield, was next to punch a hole in the visitors' rearguard but his weak shot was comfortable for Andriy Pyatov to collect. Moments later the case of deja vu all over again that would last until half-time began as United fell asleep, and the Ukrainian side came close to taking the lead. The sideways stuff that points to the concerning malaise that is inhibiting the side caused Januzaj to give the ball away. Jones chose the wrong Shakhtar runner, so Taison could slip Alex Teixeira in beyond Rio Ferdinand and after the visiting captain left the defender flailing, the finish beat David de Gea but was marginally wide of the left post.

Giggs, who joined the small band of footballers aged 40 to feature in the competition with this appearance, became the latest player in red to dawdle when he lost the ball, allowing Teixeira to again skate through and create more worry.

This term's sole shining light for Moyes has been Wayne Rooney, and the forward's 25-yard free-kick was the closest his team had come to a goal. Yet after this brushed the side-netting the alarm was again sounded as Ferdinand lost the ball to Teixeira. He surged down the left before crossing, but though Fred got off a shot the attempt was blocked.

By the break United had moved up a gear as first Rooney then Young (twice) had chances but these were spurned. The poorest came when Young pick-pocketed Darijo Srna but the winger's lob over Pyatov was spooned wide when the better option was to square for Rooney to slot into the empty goal.

This meant that 258 minutes had passed since United last scored as they embarked upon the second half. While that strike was Rooney's 57th-minute penalty in the 2-2 draw at Tottenham Hotspur at the start of the month, he could only watch as Yarolsav Rakitskyiy nearly did the trick for Mirceau Lucescu's team, though the attempt was off target.

Moyes had said the plan was to give Van Persie 20 or 30 minutes if needed, as he continues to regain match fitness following a groin problem. The manager required little time to ask the Dutchman to warm up and before he threw him on was heartened by Kagawa's volley and a Rooney sand-wedge over Pyatov: each of those was unlucky not to make it 1-0.

Then came Jones's strike. Van Persie's corner bounced once and the 21-year-old left Pyatov with no chance.

With this win United remain unbeaten in Group A and finish toplong-term absentee – their best return since the 2007-08 victorious campaign – ahead of Monday's knockout-phase draw.