United opens its Group A campaign at home to in-form Leverkusen on Tuesday, when Shakhtar eliminated in the last 16 last season travels to Sociedad.
The English champions are favored to top the group, but their start to Moyes' first season in charge has been underwhelming.
After picking up four points from its opening three games for the team's worst start to a Premier League campaign in five years, United toiled on Saturday to a 2-0 win over promoted Crystal Palace, which played with more than half of the game with 10 men.
However, Moyes did see both Robin van Persie and Wayne Rooney get on the scoresheet against Palace. And with Rooney back in action after a head injury and getting closer to full fitness, opponents should be fearful of United's front two.
"The hope is that we get a partnership where folk are saying, 'My goodness, we are having to play against Robin van Persie and Wayne Rooney,'" Moyes said. "Everybody is looking to see if those two form a partnership that can score lots of goals."
Leverkusen has never won in England in eight attempts but has started the German league season with four wins from five matches, and has a coach in Sami Hyypia who knows United well.
Hyypia spent 10 years in the Premier League with Liverpool, winning the Champions League with the club in 2005.
"We shouldn't go out with too much respect," the former Finland defender said. "I hope my players don't just look around at a stadium rich in history and get autographs from Rooney.
"They do have a new coach, but they play the same football. Man United are a great team. They won everything in the last 10 years. But to give up already before the game, I won't do that. Otherwise we could send out our second team."
Leverkusen beat United in the Champions League semifinals in 2002 on the away-goals rule, then lost to Real Madrid in the final.
Socieded has had a mixed start to the season, with only one win in the Spanish league through four rounds. It did do well to dispatch French side Lyon in the Champions League playoffs, however, to earn the club only its second appearance in the competition's group stage.
Shakhtar has lost three of its best players from last season's run to the knockout stage, with Fernandinho (Manchester City), Willian (Chelsea, via Anzhi Makhachkala) and Henrikh Mkhitaryan (Borussia Dortmund) all moving on. The team is fourth after nine games in the defense of its Ukrainian title.