Barkley's excellent equaliser and a Seamus Coleman tap-in overturned a 1-0 deficit against Norwich but, at the conclusion of a goal-laden 20 minutes, debutant Ricky van Wolfswinkel ensured the points were shared.
While he was disappointed not to claim an opening victory, however, new Everton manager Martinez was happy with his side's performance after Steven Whittaker's opener.
"It was a very good response from us," said Martinez. "We knew it was going to be difficult coming to Carrow Road with everything positive that's going on at Norwich.
"I was very pleased with the way we adapted to it. Conceding the two goals made it difficult to get the three points, but I thought we created plenty of chances and had great spells.
"We showed some good sides to our play. Clearly, it's the start of the season and we need to correct a few things, but in general terms I'm delighted."
Martinez's new playing style clearly had a big effect on his Everton side, in particular young midfielder Barkley.
The English starlet caught the eye for the Toffees with a dominating performance and impressive goal, and it did not go unnoticed by Martinez.
He added: "He's been like that throughout the pre-season. He's an enormous talent and he's a great listener.
"He wants to learn quickly, but he's also got that raw talent. He's important for the team and I'm delighted about his attitude.
"He worked extremely hard for the team and took his goal excellently."
While he was pleased with his new side's performance, Martinez did see areas to improve.
"If you look at the amount of opportunities Norwich had, we need to be better than that defensively.
"We look like a team that can create enough chances to win a game however."