The Reds overcame Ukrainian champions Shakhtar Donetsk in the final Group A fixture on Tuesday, with a second-half Phil Jones goal ensuring top spot ahead of next Monday's last-16 draw. Despite disappointing domestic results in recent weeks, United have excelled in Europe and the boss believes his team can go far in the competition, possibly all the way.
Moyes said: "I think a club like Manchester United always has to be trying to win the trophy. The teams in this competition are very good, as everyone knows, but I think you need to get better as the tournament goes on.
"Last year, I think a lot of people were surprised to see Borussia Dortmund and Bayern Munich get to the final, with no sign of the Spanish teams. I think it's a tough competition, and it will be a hard one for us, but one where we will take every game as it comes and see how we get on.
“It’s a great tournament, all United supporters love it, they’ve been to finals before and know exactly what it means. I’m looking forward to taking those fans back to another final, and hope the draw is kind to us."
Moyes was critical of the Reds' first-half performance at Old Trafford, but says his side should be pleased to qualify from a tough group.