The Spaniard's three-year tenure at Goodison Park was brought to an end on Thursday, three days before the Toffees' final match of the Barclays Premier League season at home to Norwich on Sunday.
Former Everton defender Unsworth, now manager of the Under-21s, will step into the hot-seat, assisted by his former boss at the club, Joe Royle.
The 42-year-old said: "It was a difficult day yesterday for everyone at the football club
A good man has lost his job and that's always difficult but that's football management
I think over the period you've got to win matches and it's been a difficult time for us."
Unsworth had arrived home from Everton's Finch Farm training ground when he was given the news by chairman Bill Kenwright over the phone on Thursday afternoon
"It was the proudest moment of my life yesterday when I got the call," he said
"At the same time, one of the most difficult days I've had being back at the club
"My relationship with Roberto is outstanding
He was the one who rubber-stamped me coming back to this club
I've got nothing but great words to say about him, the way he was with me and the young players
"Typical Roberto, he was the one who said to the chairman to put me in charge, so that's the measure of the man
It doesn't surprise me
He's a terrific guy."
Ronald Koeman and Frank De Boer are the early favourites for the job, although Koeman insisted at his press conference on Friday that he will see out his contract at Southampton, which has another year to run.
Unsworth threw his own hat into the ring, saying: " Who wouldn't want this job?
"I have massive aspirations to be a manager and this is the only place I want to manage
Whether it comes now, whether it comes somewhere down the line, other people will decide that
I'm just very, very proud to be given this opportunity on Sunday."
Unsworth took his first training session with the first team on Friday morning
"They responded fantastically well, we had a good session," he said
"There's not a great deal you can put into a group of players in two sessions so we tried to keep it basic and just work on a couple of things and they certainly responded to that."
Unsworth's task is to ensure Everton finish a below-par Premier League season on a high against relegated Norwich.
The Toffees sit 12th in the table after winning only one of their last 10 games, with the axe falling on Martinez following Wednesday's 3-0 loss to Sunderland
Unsworth is happy to have Royle, under whom he won the 1995 FA Cup, alongside him, joking: " It's nice that he's working for me now
"We have a great relationship, we did when I was a player both here and at Ipswich
It's been great to work with him, to tap into his knowledge and experience here for the last two years with the Under-21s
"To have him next to me, to have John Ebbrell and Duncan Ferguson, who are working with me as well, you've got four passionate Evertonians who want to see this club succeed
"We said to the players this morning that we have to finish on a high
We have to finish with a performance and a win
Evertonians have got to leave Goodison Park with a smile on their face, that's really important to us."
Unsworth confirmed Tim Howard will make a final appearance in goal before leaving the club while Phil Jagielka and Seamus Coleman could both return from hamstring injuries.
Source : PA
Source: PA