After a disastrous performance in the first leg in Greece, where United lost 2-0, the reigning Premier League champions were under pressure to turn in a much-improved showing at Old Trafford on Wednesday.
They did just that, Robin van Persie scoring a hat-trick in a 3-0 win which saw Moyes and his men through to the last eight of the competition.
United are the lowest-ranked side in their respective league in Friday's draw, but Moyes sees no reason why his side can't go on to win the trophy for a fourth time.
"I hope so," he said. "We will probably go into the quarter-final tie as underdogs but I don't see why not.
"This football club is capable of it. If we play to our capabilities, which we've not shown that often this season, I think we can be a match for any team.
"This club has history in the competition. It knows what to do at this stage.
"We're not shocked to be in the quarter-finals - it's something we do regularly. We will try to get into the last four now."
On several occasions this season United have produced a result which many believed would turn their faltering season around, but too often a positive performance was followed up by a disappointing one, and Moyes hopes that they can buck that trend this time.
"I don't want to say this is the moment but in the same breath I really hope it is," he said.
"It was a really good result in Europe as we came from 2-0 behind. We hadn't played well in the first game but we put it right.
"The result means we've got ourselves another two games in the Champions League. That's something we'll really look forward to. All I can say is that we'll try to use this result as best we can.
"I have a lot of things I need to do. It's going to take time to happen. Most importantly, tonight the players gave the crowd something after Sunday's defeat to Liverpool and I thought the crowd responded brilliantly."
