The Red Devils overturned a 2-0 deficit from the first leg of their last-16 tie to progress 3-2 on aggregate, and the Scot believes that his team could prove to be the tournament's surprise package this season.
"I suppose we'll be the underdogs in the draw, but I genuinely believe this club of is capable of doing it," The Mirror quotes Moyes as saying.
"If we show our capabilities, which we haven't done too often this season, we can be a match for any team.
"We were disappointed after the first game, losing 2-0, and it gave us a big job to turn it around, but we managed to do it in this game."
The draw for the quarter-finals of the Champions League will take place on Friday.