Besic made his first Premier League start for Everton in last Saturday's 1-1 Merseyside derby draw with Liverpool following his close-season switch from Ferencvaros.
Martinez was impressed with the 22-year-old Bosnia-Herzegovina international, and could give him another start on Thursday when Everton travel to Krasnodar in the UEFA Europa League.
The Spaniard is convinced Besic can handle the rigours of Premier League and Europa League football, and backed him to become an integral part of Everton's team in the future.
"Nothing fazes him," Martinez told the Liverpool Echo. "It's easy to forget he only turned 22 this month but he is a very mentally strong player with a lot of belief in himself.
"Mo has a big-game mentality and he's very focused. He has everything to suggest he can go on and become a big player for us.
"Against Liverpool I thought he was very controlled and not overwhelmed by the occasion in any way. For him it’s the bigger the game, the better.
"This is a young man who rose to the occasion at the World Cup when he was playing in the Maracana Stadium for his country in a high-profile game against Argentina.
"The players know they can trust him and rely on him and he has fitted very well into the squad."
'Obviously Messi is the best player in history and being compared to him means that you're doing things right, but I try to be myself': Barcelona star responds to comparisons with Argentine legend
'Darwin Nunez would be gold for Arsenal': Liverpool legend claims Gunners should have signed Uruguayan striker in January
