United have endured a miserable first season under the Scot who was hand-picked by his predecessor Alex Ferguson to succeed him when he retired after 26 years in charge at Old Trafford.
Wednesday's 3-0 victory over the Greek champions, which gave United a 3-2 aggregate success and a place in the last eight, was in marked contrast to their dismal display when they lost 3-0 at home to Liverpool in the Premier League on Sunday.
"What I am not going to say is that this result is the start of a change in our fortunes, though I hope it is," Moyes said.
"There have been a few false starts already this season and while this is a really good result and a really important win I cannot say that everything will change now.
"I have a big job to do here, a lot of changes and it will take time."
The Liverpool result left United in seventh place in the table, 18 points behind leaders Chelsea and 12 adrift of Manchester City who occupy the fourth Champions League spot.
Unless they win the Champions League, their only realistic chance of European competition next season is in the second-tier Europa League.
Wednesday's win kept them in with a chance of lifting the European Cup for the fourth time, but it appears to be a slim one with the likes of Real Madrid, Barcelona and Chelsea looking far more likely winners on current form.
Moyes was keeping his feet firmly on the ground.
"We might not be favourites to win the Champions League this season but if we go into the quarter-finals as underdogs that's fine by me," he said.
"With this squad this club can be more than a match for anyone."
The one thing that is sure is that if United had gone out to Olympiacos the pressure on Moyes would have been even greater than it has been over the last few months.
They have suffered a series of ignominious home defeats and the performance against Liverpool was probably their worst of the season.
It might not have been a vintage display against Olympiacos but it was the first time in 30 years they have overcome a two-goal first-leg deficit to win a European tie.
Striker Robin van Persie, who has often looked listless this season, showed just why United paid Arsenal £24 million for him two years ago with a coolly taken hat-trick.
United return to league action on Saturday when they visit one of their least popular grounds in West Ham's Upton Park.
A good result there and victory over neighbours Manchester City next Tuesday and perhaps Moyes might start to believe that the season really is beginning to turn around.
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