The 30-year-old Paraguay international has signed a two-year contract after passing a medical and sorting out the paperwork on his work permit.
He is reunited with ex-Wigan boss Martinez after leaving the DW Stadium when his contract expired this summer.
Alcaraz joins fellow former Latic Arouna Kone, who agreed a three-year deal yesterday, and Atletico Madrid goalkeeper Joel Robles - who spent last season on loan at Wigan - who followed Alcaraz to Goodison a few hours later to sign a five-year deal.
"We talked to a lot of clubs but I was always interested in staying in the Premier League," Alcaraz said.
"I like this competition and at this moment in my career I thank Everton for giving me this opportunity to continue in the Premier League."
On working with Martinez he added: "I know him really well and he knows me very well also.
"We had a very good relationship in the three years at Wigan and I want to continue that. Nothing is going to change, he is very professional.
"I have learned many things with him. He gave me a lot of support when I came here, when I was new in this competition.
"It is totally different and he gave me many lessons on how to adapt to this type of football.
"Thanks to him I adapted quickly to English football. There are many things I can say thank you to Roberto for, for my years here in England."
Everton's established centre-back pairing is Phil Jagielka and Sylvain Distin but if Martinez opts to introduce a three-man backline as he did at Wigan Alcaraz could slot straight in.
Even if not he is prepared to face the competition.
"There are always good players at every club. When you go to a new club you must work hard in training to get the opportunity," he said.
"Competition is always good - the more quality within the team the more you can improve yourself.
"I will try to do my best - that's the mentality I have had throughout my career.
"When a footballer changes club - whether it is at the end of his contract or a transfer - it is always like that but the idea is to give my best and try to help the team this season.
"I will try my best, work hard and we will see what happens.
"In the last three years I have played against Everton and I have played against a very strong team and a strong club.
"They have a lot of quality in the team and strong players. We always said that when we play against Everton we expect a difficult game.
"I really feel enthusiastic for the new challenge in my career. I'm looking forward to what is happening in the future."