Robin van Persie scored a hat trick in a 3-0 win over Olympiakos on Wednesday night that advanced United into the Champions League quarterfinals on 3-2 aggregate.
"I don't want to come out here and say this is the moment," Moyes said, "but at the same time I really hope it is. We know it is a work in progress and it is going to take time to get things the way we want it."
Borussia Dortmund, defeated by German rival Bayern Munich in last year's final, lost 2-1 at home to Zenit St. Petersburg in the night's other match but advanced on 5-4 aggregate.
Van Persie converted a penalty kick in the 25th minute, side-footed in his second goal in first-half injury time and curled a free kick past goalkeeper Roberto in the 51st. Van Persie was carried off on a stretcher in the final minutes at Old Trafford after hurting his left leg in a challenge. He seemed to be in pain, holding his face and slamming the ground with his hand.
United earned the penalty kick when van Persie received Ryan Giggs' cross-field pass and crumbled when hit in the back by Jose Holebas. Giggs sparked the second goal when he floated a pass to Wayne Rooney, who crossed for Van Persie to finish. Roberto appeared to be screened by his defensive wall when van Persie curved in the free kick for fourth Champions League goal this season and 14th overall.
"You see tonight, I'm still enjoying my football," said 40-year-old Giggs, who has not committed to playing another season.
With Manchester United seventh in the Premier League, winning the Champions League likely is the only chance the team has to qualify for next year's edition of Europe's top club tournament.
Atletico Madrid, Barcelona, Bayern, Chelsea, Paris Saint-Germain and Real Madrid also are in the quarterfinal draw Friday.
"We probably now will go into it as underdogs," said Moyes, the United manager. "Hopefully, we can do it, I don't see any reason why not. If we play to our capabilities, which we have not done too often, then I think we will be a match for any team."
Olympiakos dominated the final 25 minutes, but David De Gea made several fine saves and centre backs Phil Jones and Rio Ferdinand made timely tackles.
"We had chances. We were unlucky, and we're pretty upset about it," Olympiakos midfielder David Fuster said.
At Dortmund, Hulk put Zenit ahead in the 16th minute, Sebastian Kehl tied the score in the 38th and substitute Jose Rondon got the go-ahead goal in the 73rd. Dortmund forward Robert Lewandowski will be suspended for the first leg of the quarterfinals after getting a yellow card for a hand ball, his third caution of the competition.
"The yellow card was laughable," Dortmund coach Juergen Klopp said.