After Aaron Lennon had been fouled by David Jones early on at Goodison Park on Saturday, Barkley stepped up to take the resulting spot-kick, despite set-piece specialist Leighton Baines being in the starting line-up.
The midfielder's tame effort was saved comfortably by Tom Heaton, although Everton's blushes were spared 17 minutes later when Kevin Mirallas fired home the winner.
Martinez, who added that Romelu Lukaku – on the bench at the time – is Everton's current first-choice penalty taker, claimed that Barkley's performance after the miss showed what a top player he will go on to become.
"Ross showed responsibility and felt good in taking it and Leighton [Baines] was happy for him to do so," said the Spaniard.
"Heaton deserves credit for the save, but I love seeing players taking control of the big moments and showing that character – even though Ross missed, he proved he's a top player.
"It shows the potential that Ross has and he showed me that he's an incredible footballer with a high level of maturity that's going to become an Everton great."
The hosts controlled proceedings from the off on Saturday, with Sean Dyche's relegation-battling Burnley struggling to cope with Everton's pacey front line.
Though the penalty was missed, Mirallas made no such mistake just prior to the half-hour mark, prodding home from close range after latching onto Seamus Coleman's cross.
To make matters worse for the visitors, Ashley Barnes was shown a second yellow card on the stroke of half-time, although Mirallas was lucky to escape with just a booking for a lunge on George Boyd after the interval.
However, Martinez, who was delighted with his side's efficient performance – their fourth win in five Premier League games – insisted that there was no malice in the Belgian's tackle.
"I don't think it [the Mirallas challenge] was a red card, it's not a malicious tackle from Kevin, it was just a normal striker's challenge," he added.
"There were a few difficult decisions for the referee, but I don't think the result was down to them."
