The Toffees struggled in the first half at Goodison Park after Aiden McGeady hit a post and trailed to Leandro Bacuna's 34th-minute opener.
But Everton were a different prospect after the break and pulled level through substitute Steven Naismith 16 minutes from time before Kevin Mirallas earned a 2-1 win with a spectacular free-kick.
The victory, coupled with dropped points for Tottenham and Manchester United, repaired some of the damage from Tuesday's Merseryside derby hammering by Liverpool.
Everton climbed back to fifth place and to within a point of Liverpool, albeit having played a game more.
Manager Martinez said: "From our point of view it's the most satisfactory victory of the season.
"You could see in the first half we were very cagey and wary and that stopped us getting on the ball and having the fluency that we normally have.
"When you hit the post and then you concede a goal, it becomes a real test of character, and I thought the way we reacted in the second half was one of the most pleasing aspects that you can have as a manager.
"With the character and the talent and the desire in that dressing room, I couldn't be prouder. It's a moment to get really excited about the end of the season after that sort of performance."
Martinez denied their first-half display was a hangover from the 4-0 defeat at Anfield, blaming tactical changes he tried to implement.
The Spaniard said: "It was my fault more than anything. In the first half we were too responsible. There were too many players who wanted to do the right things without enjoying themselves.
"And we are a team that when we enjoy ourselves we are quite difficult to play against. I think that's what we proved in the second half."
In the absence of Romelu Lukaku and new signing Lacina Traore, Mirallas took over striking duties, and it was generally not a happy experience for the Belgian.
But he had the last word, picking himself up after being fouled by Ciaran Clark five minutes from time and placing the ball beyond the reach of Brad Guzan from 30 yards.
Leighton Baines has made such strikes his speciality but the England man had a watching brief this time.
Martinez said: "Kevin found it very frustrating, as you would expect. He was surrounded by Aston Villa players and we couldn't get into the final third.
"They were trying to stop us really high up the pitch. Kevin was working extremely hard without getting any sort of joy.
"I think Kevin Mirallas' quality is as good as Leighton Baines, obviously with different feet. It would be difficult to separate them.
"What I like about the two players, more than the technical aspect, it's the meaning of scoring a goal to get three points. They've both got that mentality of controlling the moment and that's something you need to be born with."
Villa went into the match on the back of a very different derby experience after beating West Brom 4-3 on Wednesday, and manager Paul Lambert felt they deserved something at Goodison.
He said: "The first goal was really poor from our point of view. I think the second goal you hold your hands up and say it's a world-class free-kick, and you can't do anything about that.
"I don't think we deserved to lose the game. We took the sting out of the crowd and frustrated Everton, who are a good side.
"In the first half, other than Aiden's shot against the post, there's not a save that Guzan's had to make, and we scored a really good goal."