The England international wore the armband several times in place of Phil Neville last season, and the newly appointed boss has confirmed he will keep the role.
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Martinez told the club's official website: "I’ve been a big admirer of Phil Jagielka and I do feel that now he is probably at the peak of his career in terms of the leadership he has achieved when representing England and the roles that he has had for the national team and for a few seasons now at Everton.
"It's the right moment for him to take that role and, for me, it is representative of the standards of the standards of our football club on a daily basis."
The 30-year-old has been a consistent performer for Everton and England in recent years, and Martinez believes he is the perfect man to lead the side as they move on from Moyes' tenure.
"I know that he is going to embrace that role and be a perfect fit for a role that is traditionally a great honour at the football club," he added
"Phil Jagielka is going to be a phenomenal captain."