Under performing and under fire in the Premier League, David Moyes' men responded to their first consecutive home league defeats since 2002 with a battling display on their return to the top table of European football to progress to the knockout stages on a high.
Phil Jones' 67th-minute half volley ultimately proved the difference, but Shakhtar were left to rue what might have been as they controlled the action for long periods.
The Ukrainians failed to convert their chances - most notably Brazilian Alex Teixeira - as Bayer Leverkusen's 1-0 win at Real Sociedad sent them out.
United started with Robin van Persie on the bench, as Wayne Rooney was asked to lead the line alone.
That provided another chance for Shinji Kagawa to stake his claim playing off the front, as he returned from the illness that had bizarrely required him to have his stomach pumped due to over-eating as he missed the Premier League loss to Newcastle.
Shakhtar coach Mircea Lucescu handed returns to Brazilian strike duo Fred and Taison but it was United who started more brightly, as Jones forced goalkeeper Andriy Pyatov to block.
The visitors should have taken a 12th-minute lead when a defence-splitting ball from midfield set Teixeira away, but he dragged his shot wide.
At the other end, Adnan Januzaj controlled expertly on his chest before spinning and firing goalward, forcing a fine save from Pyatov.
Shakhtar were the more dangerous in the final third – Teixeira winding his way past several challenges midway through the half with a run that deserved better than his tame finish.
However, Ashley Young wasted two gilt-edged chances to give United an advantage as first he outmuscled Darijo Srna, only to lob beyond the far post, before slicing woefully wide.
There was no doubt some strong words from Moyes in the dressing room, but the home side seemed to still be taking them in when a corner across the box missed everyone a minute after the restart and Yaroslav Rakitskiy scuffed a shot against the outside of the post.
Finally, United burst into life as Rooney's audacious chip from distance almost caught Pyatov off his line.
Reinforcements arrived in the shape of Van Persie and Tom Cleverley, and the Dutchman had not even touched the ball when Rooney's clever pass almost played him in.
But it was Jones who stepped up when United needed a hero.
After they had pressed Shakhtar to win a 67th-minute corner, Van Persie's flag-kick from the right bounced through everyone in the box where Jones angled his body to drill beyond Pyatov.
Srna forced David de Gea into a smart save late on, but United held firm and Van Persie almost added late gloss to the scoreline when a clipped finish flashed wide.
