It was former boss David Moyes who decided to hand Jagielka the armband after Phil Neville announced that he would be leaving the club in the summer.
Moyes has also moved on to replace Sir Alex Ferguson in charge of Manchester United, but Martinez agrees that Jagielka is perfect for the role.
The Spaniard, who has signed a four-year deal at Goodison Park, says Jagielka has the leadership qualities to succeed as skipper and will also command the respect of his team-mates with his performances on the pitch.
"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," Martinez told evertontv.
"It's the right moment for him to take that role and, for me, it is representative of the standards of our football club on a daily basis."
"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.
"Phil Jagielka is going to be a phenomenal captain."