Phil Jones' second-half volley secured a nervy 1-0 win over Shakhtar Donetsk at Old Trafford.
The result sealed top spot in Group A and ensured United will avoid Real Madrid, Bayern Munich, Paris St Germain and, almost certainly, Barcelona when the last-16 draw is made next Monday.
After back-to-back home defeats, it was at least the result United craved.
However, 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."
Moyes claimed there was little difference between Tuesday night's performance and those in the defeats to Everton and Newcastle that had left United on the brink of suffering three successive reverses at Old Trafford since 1962.
Such a scenario would have increased the air of crisis around Old Trafford and Alex Teixeira, Fred and Vyacheslav Shevchuk all had a chance to put the visitors ahead before Jones lashed home Robin van Persie's volley.
Yet United ended their campaign unbeaten - having 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 serious 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.
"But you want to try and win the group. A lot has been said about me being inexperienced in the Champions League so I am happy to top the group."
Moyes confirmed Nemanja Vidic, Patrice Evra, Marouane Fellaini and Chris Smalling had all missed out through injury, although added none of the problems were serious.
Now United must get their domestic form back on track at Aston Villa on Sunday.
And in that, they have the good wishes of Shakhtar coach Mircea Lucescu, who cast aside the disappointment of his side sliding into the Europa League to give Moyes his backing.
"I would like to congratulate Manchester United on their victory and wish them good luck for future games," said the Romanian.
"I hope this result will help them to revive their play in the Premier League."