United lost the first leg of their last-16 tie at Olympiacos 2-0, but they claimed a 3-2 aggregate win following a night of high drama at Old Trafford.
Robin van Persie scored his first hat-trick under Moyes to achieve a 3-0 win which puts United in the hat for Friday's draw.
Given their dismal domestic record, United are big outsiders for the competition, but Moyes still believes his team can defy the odds and make it to the final.
When asked whether his team could now become dark horses for the trophy, just as Chelsea were in 2012 and Liverpool were seven years earlier, the United boss said: "We probably now will go in as underdogs in a lot of the games.
"Hopefully we can do it, I don't see any reason why not. I think this club is capable of it.
"If we play to our capabilities, which we have not done too often, then I think we will be a match for any team."
Van Persie put United ahead from the penalty spot in the 25th minute after he was bundled over in the box by Jose Holebas.
The Dutchman then converted Wayne Rooney's pass to make it 2-0 before sealing the win with a 25-yard free-kick.
Moyes could not speak highly enough of the player's performance afterwards.
"To score a hat-trick in Champions League football is a big thing," the Scot said.
"I think there are only certain players in the world capable of it, Robin van Persie is one of them.
"He gets his penalty kick, he scored a great free-kick and Wayne did brilliantly well for him for the second goal. It is great that he got a hat-trick because he is a top player, he really is, and I think he showed that tonight."
Moyes cut a worried figure when Van Persie was carried off on a stretcher in the dying minutes following a tackle by Giannis Maniatis.
But the United boss was confident the injury sustained by the former Arsenal striker was not serious.
"I don't think he is too bad," Moyes said.
"He has got a knock just behind his knee. We will assess it - we will know more in the morning.
"He has got a knee in the back of his knee but hopefully it is not too bad."
Moyes also found time to praise 40-year-old Ryan Giggs, who played a key role in the build-up to the first two goals.
"I thought Giggsy was fantastic," Moyes said.
"The passes he made for the two goals, his general play - more importantly his general fitness... Sometimes you think he'll need to come off after 60 minutes but he's a freak. He's something different.
"It was a big night, we needed it. It was off the back of a bad result and I thought we needed his experience and he showed it."
United fans chanted Moyes' name regularly throughout the second half - something which did not happen on Sunday when the Red Devils lost 3-0 to Liverpool.
That defeat - the latest in a dismal domestic season for the club - led to questions over Moyes' position, but he denied he felt any pressure that he might lose his job if the result did not go his way on Wednesday night.
"I don't think so," he said. "I see myself here for a long time.
"I'm not feeling any pressure from inside the club. It's all coming from you people. Everybody is fine, we know the job I have to do. It's a big job, probably bigger than we all expected when I first came in.
"The players have been great. I keep hearing that this one has fallen out, or someone else has fallen out. I keep telling people it is rubbish and it is rubbish.
"People are looking for reasons we've not done well. The reason we've not done well is we've not played well enough in games and that's the truth behind it all."
Olympiacos did not put up anywhere near as big a fight as they did in the first leg, although things could have been different had David De Gea not pulled off two outstanding saves in quick succession just before United went 2-0 up.
Olympiacos manager Michel denied that his team froze under the pressure of playing in front of a 75,000 plus crowd at Old Trafford.
"I don't think they were scared. Yes, we made some errors but these errors were not the result of fear, just a lack of concentration," the former Real Madrid midfielder said.
"We were determined to play our football, but football sometimes doesn't go the way you plan."