After a first half of few opportunities, the game burst in to life six minutes after the interval as Norwich took the lead against the run of play when Steven Whittaker reacted quickest to convert the rebound from his initial shot.
Barkley was undoubtedly Everton's best player as he put in an all-conquering performance, leading Martinez to praise the playmaker.
"We are very pleased with his overall contribution," the former Wigan boss said.
"Tactically he was outstanding. That was a very mature performance for a young man. I was very pleased with that because we've seen that in pre-season.
"You want to have as many players who can give you something special as you can. Ross has got that talent. I think we have a few throughout the squad. And that's important for competition.
"I'm pleased with the manner in which he wants to learn and to listen and wants to develop. To have a player with that ability at that age is an incredible asset for us."
Barkley's progression through the ranks at Goodison Park has stuttered in recent years due to a catalogue of injuries but Martinez believes his opening-day display proves he is prepared for an influential campaign.
"He's been at Everton for a long time and he's had his problems with injuries," Martinez added.
"I can only speak about his attitude and the way he looks after himself and the way he works in pre-season. I'm pleased he's been able to show everyone he's ready.
"That's what is important about his play. He can understand what he needs to do to be effective off the ball. He wants to take information in and wants to understand. I thought he was perfect at it."
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- Ricky Van Wolfswinkel
- Seamus Coleman