Centre-half Stones had a transfer request rejected earlier in the week amid reports of a third bid from Chelsea, who have offered upwards of £30million for the England international.
Martinez saw fit to hand the 21-year-old a start at White Hart Lane despite fears over his reception from the away fans, and Stones helped his side to a clean sheet.
Instead of struggling under the pressure of such a publicised interest from Chelsea, Martinez believes the former Barnsley man “has grown” from the experience.
“John Stones was an immense influence on what we did defensively,” he told reporters. “The reading of the game, he looks like a player who has grown in the last few weeks under the pressure and speculation.
“Every single Evertonian will show that respect to him for the way he has handled the situation. You can see the signs of a future England captain.”
On the game, Martinez added: “I thought it was a really satisfying performance. It was a different type of game to adapt to. We had to find character.
“At spells we looked like we didn’t have that sharpness in posession that we normally do.
“They looked a little bit fresher, but overall I couldn’t be happier. We had to defend well and Tim Howard was exceptional.
“To come here and perform in the manner we did was satisfying. It’s a sign of what we have.”
Midfielder Tom Cleverley was substituted late in the first half after crumbling under an innocuous challenge from Eric Dier, but Martinez allayed fears over the fitness of the ex-Manchester United star.
“I think the doctors are a little more positive now. Straight away we felt like it could have been something else.
“To be able to get through that with only a minor injury has us thanking the luck we had.”