While crediting Villa for giving his team a tough match, Martinez believes that his players composed themselves well throughout.
"It was very difficult. Villa carried a threat on the counter-attack. I'm pleased we controlled the game and stopped them. We scored at the right times and we were in control, but I'm so pleased at the way we performed," he is quoted as saying by BBC Sport.
"We had a real focus and intensity to defend well. Overall, we worked extremely hard to keep Villa quiet in front of goal. It was a very clever performance and an important one. It's a special feeling after the way we lost at Manchester United."
Everton turn their attention back to their Europa League campaign on Thursday when they face Lille in France.