Mata confirmed Chelsea owner Roman Abramovich had spoken to the squad in the aftermath of relieving Villas-Boas of his duties on Sunday and is demanding an improvement in results.
The Blues began that process with a 2-0 win at Birmingham on Tuesday night to reach the last eight of the FA Cup with Mata scoring the opening goal to set up a home tie with Leicester.
When asked if the players are now playing for their futures, Mata said: "I don't know if the players are playing for their future. But we have to prove ourselves every game.
"For me and the other players, we are playing in a top squad and for a top club - and we must always do our best in every game.
"Roman (Abramovich) told us that we have an opportunity to change this run of defeats. Against Birmingham, we did so we have to keep going and try to get better."
Mata insists the Chelsea squad are supportive of caretaker boss Roberto Di Matteo and claims they were "sad" to see Villas-Boas leave Stamford Bridge.
He said: "We all knew Robbie from before. He is a very good guy as well, he knows Chelsea from the inside, he used to play here. We are all with the new manager because it is what we have to do.
"This week was not easy for us with a change of the manager but, when you're on the pitch, you're always thinking to give your best to try to win and that's what we did.
"It's never good news to change a manager. If this happens it's because things have not been good. Andre was a very good manager and a very good person as well. We are very grateful to him.
"We were sad when he left because it's always a difficult moment when a club changes its manager, but now we are going to keep working without him and with Robbie instead."
Mata is aware what will be regarded as a relatively successful season after all the alleged in-fighting.
He said: "We are going to keep giving 100% for the rest of the season to achieve our objectives.
"We will try to win this FA Cup and against Napoli in the Champions League I think we can do it, it is possible for us to win 2-0 and over-turn the first leg.
"The most important thing to think about is the coach we have now, Robbie Di Matteo. We must just think about this season and finish the next two months well."
On the downside Fernando Torres drew another blank, has now gone 23 games without a goal since October and even appeared to pass up the opportunity to take a penalty.
Mata took the spot-kick himself although it was saved, as in the first meeting, by Birmingham keeper Colin Doyle.
Di Matteo said: "Fernando had a great game, fantastic game. Yes, he missed a chance but other players missed chances too. He had a fantastic game.
"In terms of the penalty, to be honest, that's the players' decision on the pitch.
"I was far away. We had two penalty takers but the players have to take that decision."
Di Matteo admits he did not need much persuading to remain with the club he played for with distinction and steer them through to the end of the season.
He said: "I'm a Chelsea boy and they just said we were going to stay until the end of the season and try to win as many games as possible with the rest of the staff.
"That's it really. I didn't really think too much about it."
The Chelsea fans also chanted Di Matteo's name in the second period after initial support for ex boss Jose Mourinho.
Di Matteo said: "They love the club and want to see a team that shows the passion, effort and commitment.
"You can then lose a game but we need to take those things into games. I've been many years here and I've always had a very good connection with then.
"Many players have been here a long time and we're all in the same situation, the same boat really."