United are looking to rescue their season by overturning a 2-0 first-leg deficit in their Champions League last-16 clash.
The reigning champions have no chance of defending the Premier League title this season and are now unlikely to finish in the top four, and after crashing out of both domestic cup competitions the Champions League provides a last chance of glory.
But a sluggish performance in Athens has left United trailing to goals from Alejandro Dominguez and on-loan Arsenal striker Joel Campbell.
Only five teams have overcome a deficit of two goals or more in the last 22 years in the competition, but Moyes is not giving up.
"If we can go through it would be a massive lift for us," the United boss said.
"We know we have got ourselves in a poor position being 2-0 down but we have a lot of belief and we have to make it show in the game.
"We need to start giving something back now. We need to start performing together, me, the team. We need to make sure we work together to get a performance and a night to be remembered.
"And we'll do everything we possibly can to make that happen. We'll leave nothing behind on the night, hopefully, and we'll try and make sure we somehow get through to the next round."
Meanwhile United full-back Patrice Evra is determined to end the season on a high.
The France international is out of contract at Old Trafford in the summer, but he insists his only focus is on getting a positive result against Olympiakos.
Evra said: "When I put on the Man United shirt I will give everything, no matter if I will stay or if I will leave.
"That's what is most important tonight. I'm more focused about giving everything because I really want to give a good performance and make sure Man United qualify. That's my only worry tonight.
"I think everyone wants to fight for this club, everyone who loves this club. We know we had a bad game in the first leg.
"I can promise you we're going to all fight and respect the shirt and make sure all the fans will be really proud of us after the game."
Manchester United v Olympiakos is live on Sky Sports 1 and 1HD from 7pm