Barkley, who made his full England debut in the 4-0 World Cup qualifying win over Moldova in September, turned in another impressive display on Monday night and Martinez predicts a bright future for the 19-year-old.
"You are looking at the way he is growing game-by-game, he is becoming stronger and bigger. He has a terrific understanding of the game at such a young age," he told the club's official website.
"It's really pleasing and nobody deserved the goal more than Ross because of the work that he has put in in the previous five games.
"It was a special moment. Ross is a young diamond of English football. There is a long way to go, but he knows that. He wants to work so hard and that is a pleasure to see.
"I'm so pleased for him. To hit the back of the net for the first time at Goodison is going to be a special memory."