The 39-year-old takes over from David Moyes, who is due to succeed Alex Ferguson as manager of Manchester United, after being granted permission to leave Wigan Athletic by chairman Dave Whelan.
Despite failing to win a trophy in his 11-year tenure at Goodison Park, Moyes honed Everton into a side who regularly punched above their weight.
They finished sixth in the Premier League this season, which was their seventh consecutive top-eight finish, but Martinez said he was not daunted by the prospect of following in the Scot's footsteps.
"There is huge pressure, but I am extremely proud of that pressure because it means what David Moyes did in 11 years is set high standards and lay an incredible platform," he said.
"I feel lucky to be able to carry on that work. He has given Everton an identity and an incredible standard and all I want to do is be humble and hard-working enough to take this magnificent club to the next level."
He added: "Finishing in the top six was magnificent for Everton, but we need to achieve more. The aim for Everton should be to get into the Champions League. That doesn't mean next season, but that has to be the aim for the future.
"It is not going to happen overnight. In football you need to be creative. We want to be involved in Europe and the Champions League, and we have to have the dream to fight for."
Martinez admitted it had been a wrench to leave Wigan, where he previously spent six years as a player, but said the club were well placed to bounce back from relegation to the Championship at the first attempt.
"I believe in human relationships. It is never easy to break them," he said. "Hopefully, I can build them at Everton and be a really strong unit.
"I believe Wigan have got a great future. It is the right time for a fresh energy around the place."
Everton chairman Bill Kenwright, who reportedly also held talks with Porto coach Vitor Pereira and former Schalke coach Ralf Rangnick, said he was convinced he had found the right man.
"When David (Moyes) first came to see me, he sat down 11 years ago and we were in a bad state, and he said 'We're not going down,'" said Kenwright.
"Almost Roberto's first words to me were: 'I'll get you in the Champions League.'
"To the Evertonians: if you'd been at the meetings with our new manager, you would be as thrilled, as excited, as buoyed and as in awe as I have been."
Kenwright added that Martinez would be given "a few quid (pounds)" to sign new players and said he was "confident" the current squad would remain intact.
British media reports said Everton had agreed to pay Wigan £1.5 million to release Martinez from the final year of his contract.
The amiable and erudite Spaniard led Wigan to a shock victory over Manchester City in this season's FA Cup final -- the club's first major honour - but they were subsequently relegated from the Premier League.
He has earned a reputation for producing teams that play attractive, passing football during his spells with Wigan and previous side Swansea City, and Whelan tipped him to be a success at Everton.
"He's a wonderful guy, very honest and he gives you 150 per cent, totally," Whelan told Sky Sports News. "We're sorry to lose him, but I'm delighted he has gone to Everton. I wish him the very best of luck."