Chairman Bill Kenwright revealed Martinez's ambitions and confirmed that funds will be available for the former Wigan boss to try and realise them.
"When David (Moyes) first came to see me, he sat down, and we were in a bad state. First words, he said 'we are not going down'," Kenwright said.
"Roberto's first words, he will get us in the Champions League. He had obviously studied the football club and recognised what David Moyes has achieved.
"He understands the game and Everton. He told us about our history and what he wanted to do. It was a terrific meeting. All I can tell you is, when you know, you know.
"I'll give him a few quid to sign a few players. We all the situation at Everton, we all know that there are players with release clauses but they are not for sale.
"I have listened to the fans for the last four weeks and their voices have come round loud and clear. 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."
The 39-year-old is itching to begin his tenure at the Merseyside club, but concedes he will face immediate pressure to build on the work of his predecessor.
"There's not much I want to say apart from it's a really special day to come to Everton Football Club," Martinez said.
"I have this feeling already of excitement and honour, and I would like to thank the chairman for giving me the opportunity to come into this football club.
"There's huge pressure, but I'm extremely proud of that pressure because that means what David Moyes has done at Everton in the last 11 years had set a huge platform for the next manager.
"What he's done is given Everton an identity and incredible standards. All I want to do is be humble enough and hard-working enough to take that to a new level. What David Moyes has done is set a real high standard."
Martinez won plaudits after guiding Wigan to an unlikely FA Cup success against Manchester City last season, but feels the time is right to hand over to a new man at the DW Stadium.
"It's been a phenomenal time at Wigan. After four seasons, this was the right time," he added.
"The circumstances at the club are different now, they are going to be playing in Europe and trying to get out of the Championship as soon as possible.
"I felt it was time for a new man to take that on board. I wanted to look for a new challenge. From that moment on it became clear that the only club I would fit in at was Everton.
"After I met the chairman, I knew Everton was the right football club. It is the right time to be involved in such an exciting challenge."