And the Spaniard insists the players will give everything to forget the crushing 3-0 defeat at home to arch-rivals Liverpool, starting on Wednesday when the Greek side arrive at Old Trafford holding a 2-0 lead from the first leg.
Manager David Moyes admitted in the aftermath of a fifth home defeat of the season on Sunday that, from what he had seen in the build up, he had no reason to expect the level of performance he got from his players.
Mata, who is ineligible to face Olympiacos having featured in the competition for Chelsea prior to his £38million January transfer, was in agreement.
"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 Olympiacos 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 help."
Mata had nothing but praise for United's home support and the way they reacted to defeat against the Reds.
The 25-year-old admits he was "mad" they were not able to give fans what they deserved.
"There are no words to describe your support in the stadium," he added.
"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."
Mata believes the experience will make him a better player.
"The storm will pass and the sun will rise again. I have no doubt," he said.
"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.
"And when all this is gone I'm sure I will be a more mature footballer."
It does not get any easier for Moyes, however, as Everton academy coach and former player Kevin Sheedy has put the boot into the Scot.
Sheedy has been on the Toffees' coaching staff since 2006, and is currently in charge of the under-18s, so has first-hand experience of working under Moyes before he moved to Old Trafford last summer.
However, he was far from complimentary after Sunday's defeat.
"All of you out there Moyes was never interested in our youth team or players," Sheedy, who played 314 times for Everton from 1982 to 1992, tweeted after the Red Devils had been beaten by a Steven Gerrard brace of penalties and a goal from Luis Suarez.
"In my 7 years moyes showed no intereset [sic] in our youth team."
Asked on the social media site whether that must have "done his head in", Sheedy replied: "Yep, he did.".
Press Association Sport understands when Everton became aware of those tweets they asked Sheedy to remove them and they were subsequently deleted.
However, his criticism of Moyes' tactics - "Punt the ball up to Fellaini. Great viewing," - in Sunday's defeat remained.
In response to another follower Sheedy, a former Republic of Ireland international, wrote: "We now have a Manager who wants to win games. COYB."