Reports had suggested that the Old Trafford hierarchy were beginning to lose faith in the United boss after a difficult season, with the 3-0 Premier League humbling against Liverpool at the weekend only increasing the speculation surrounding the Scot.
But Robin van Persie helped lift the gloom on Wednesday with a hat-trick as United completed a remarkable comeback and moved into the quarter-finals of the Champions League with a 3-0 win on the night to secure a 3-2 aggregate win.
The relief among the home supporters at the final whistle was obvious, but Moyes shrugged off any suggestions that the last 16 clash was a pivotal point in his managerial career.
"People will probably think turning points in people's careers, but I didn't see it that way because I know what the club have got here and the club know what they've got here, and I see myself being here for a long time," he told Sky Sports.
"I'm not feeling any pressure from inside the club. Everybody's fine, we know exactly the job we have to do.
"It's a big job, probably bigger than maybe we all expected when I first came in, but the players have been great. I keep hearing things about 'this one's fallen out' or 'someone else has fallen out' and I keep telling people it's rubbish. People are trying to look for reasons why we're not doing well - the reasons are we've just not played well enough in the games and that's the truth behind it all."
The Scot went on to express his delight with the manner of the performance as United overturned their two-goal first-leg deficit, particularly in light of the disappointing loss to Liverpool.
"The way the players responded after Sunday's defeat - I did say on Monday morning we have to win 3-0 and give the crowd something back, because the crowd here have been incredible, they really have," he continued.
"Even tonight, they were fantastic, so they deserved a good night because we've not given them many good nights this season.
"The players were incredibly committed. I think they're hurting as well. They know we're not doing as well as we should and we certainly didn't play well in Olympiacos.
"But tonight they put it right, we deserved it."
Moyes reserved special praise for 40-year-old Ryan Giggs, who completed the full 90 minutes and played a major part in the build-up to two of United's goals.
"I thought Giggsy was fantastic, he really was," he added. "The passes he made for the two goals, his general play and, more importantly, his general fitness.
"Sometimes you're thinking he might need to come off after 60 minutes, but he's a freak, he really is, he's something different.
"It was a big night, we needed it, we were off the back of a bad result, I thought we needed his experience and he certainly showed it."
Meanwhile, Moyes played down fears after Van Persie was carried off on a stretcher late on, insisting that his knee injury was "not that bad".
