The Red Devils remain unbeaten, have won all their home matches and conceded just three goals, the lowest of any team in the four completed groups.
Their form is in stark contrast to the awful results in the Premier League, where United have slithered to ninth and look incapable of mounting a significant defence of their championship.
And Moyes cannot adequately explain why.
"If I had the answer I would have solved it by now and it would all be okay," he said. "I do believe there is a different style and you can tell the players here are used to it.
"They are well experienced in European football so I think they are more than capable of playing at this level. They are comfortable with it.
"They are also very comfortable in the Premier League but at the moment they are not showing it."
After back-to-back home defeats, Phil Jones lashed home Robin van Persie’s second-half corner to secure the result United craved.
Yet the performance was unconvincing at times – Shakhtar had nine shots on target to United’s three – and Moyes knows much more will be required when his side resume their campaign in the spring.
"We all know we as a team have to improve if we are to keep progressing in the competition," said Moyes.
"We didn’t play particularly well in the opening 30 minutes. We could have been fortunate still to be 0-0 but we played much better in the second half."
It might not be enough to stave all United’s critics off either, given Roy Keane branded Ashley Young as "not good enough" in his half-time assessment of the contest.
While the Irishman’s repeated assaults on his former club have the edge of bitterness, there are many United supporters who agree.
However, Young was amongst those singled out for praise by Shakhtar coach Mircea Lucescu.
"Manchester United is a very strong team," said the Romanian. "I would like to single out Rafael, Jones, Young, (Adnan) Januzaj and (Alexander) Buttner.
"But Wayne Rooney also played with enthusiasm. You could see his desire and will to win. I wish Manchester United good luck for future games and hope this result will help them to revive their play in the Premier League."