United's season hangs on the result, with the side having lost 2-0 in the first leg in Greece and suffered a morale-sapping 3-0 defeat by bitter rivals Liverpool in the Premier League on Sunday.
That left the English champions in seventh place and unlikely to qualify for next term's Champions League unless they win the tournament this season.
Mata, who is ineligible to face Olympiakos having played in the Champions League for Chelsea prior to his Â£37.1 million (44.7m euros) transfer between the Premier League rivals, insisted United could bounce back.
"I have to say that our week at the training ground had been good and we were very hopeful ahead of this derby," he wrote in his weekly online blog.
"However, the day of the match everything went bad. It was a tough defeat and I want to tell you that we will give everything we have in order to forget about this in the remaining games.
"And there is a very important one around the corner.
"We know the history of this club and its achievements, based on a winning spirit.
"That is what we need to beat Olympiakos and get through in the Champions League.
"As you know I cannot play in the Champions League any more this season but I'll be backing the team in Old Trafford as any other fan.
"If we score in the first few minutes I think we can make it with your (the supporters) help."
Mata had nothing but praise for United's home support and the way they reacted to defeat against the Reds.
Mata, 25, thanked United fans for their backing against Liverpool.
"There are no words to describe your support in the stadium," he said. "In games like yesterday it makes me mad not being able to give you what you deserve.
"I know there is nothing I can say right now but at least I want you to know how I feel."
However, he added: "The storm will pass and the sun will rise again. I have no doubt.
"Besides, no one said this would be easy but this is football. It gives you fantastic moments but also very hard times you have to cope with, when you have to show pride and professionalism until the end."
(For more sports news and updates, follow Ahram Online Sports on Twitter at @AO_Sports and on Facebook at AhramOnlineSports.)