The 19-year-old chipped the vital penalty as the Toffees triumphed over the Serie A champion 6-5 in a shootout, and will now face either Real Madrid or the LA Galaxy in the ICC.
The Premier League side defeated the Serie A champion 6-5 on penalties after a 1-1 draw in the Guinness International Champions Cup in San Francisco.
And Stones, who moved to Goodison Park from Barnsley in January, stole the show in the decisive shootout when he converted a cheeky chipped penalty after Andrea Pirlo had sent his kick wide for Juventus.
Martinez revealed he was not taken aback by the 19-year-old's choice of spot kick, and he praised his calm and collected approach in such pressurized circumstances.
"I knew he was going to do it, because the last three he has taken in training he has done that. The question was could he do it after Andrea Pirlo and at his age," Martinez told the club's official website. "It was incredible and I thought it was an example of his performance. He was composed throughout and it was a very useful game for himself and [he could be] a very important 'new' player for us going forward."
Stones also helped to nullify the Antonio Conte's dangerous new-look attack of Carlos Tevez and Fernando Llorente during the game, and he will now get the opportunity to pit himself against either the LA Galaxy or Real Madrid in the next round of the tournament at Dodger Stadium on Saturday.
The friendly match -- which was sealed with a Tim Howard penalty save -- proved a good test in the Everton's build-up to the new Premier League season, and Martinez was pleased with the outing.
"It was really, really useful," Martinez said. "The effort from the players and the understanding to go toe-to-toe with such a good side was a really good exercise."