The Toffees boss believes his side eventually did enough in the game to deserve to take all three points, though he did take the blame for a lacklustre first-half display
Kevin Mirallas' late free kick wrapped up the three points after a much improved second half showing saw Steven Naismith cancel out Leandro Bacuna's opener, and Martinez was proud of the way his side fought their way back into the match.
"I thought it was, from our point of view, the most satisfying victory of the season," he told reporters.
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"When you get a situation where you hit the post then concede at the other end, it becomes a real test of character, and I thought the way that we reacted in the second half was one of the most pleasing aspects that you can have as a manager. I couldn't be prouder."
Mirallas' goal was the obvious highlight on what was a frustrating afternoon for the Belgium international as he replaced Romelu Lukaku as the lone striker, but Martinez was pleased with the way the 26-year-old reacted to seal victory.
"Kevin found it very frustrating. He was surrounded by Aston Villa players and he couldn't get into the final third.
"You could see by the way Aston Villa set up that they were trying to stop him high up the pitch, and Kevin was working extremely hard without getting any joy.
"But the way he took that free kick showed he is ready for those moments where he can take control of the situation of the game, and that is shown by the calibre of the player."
Meanwhile Paul Lambert was also proud of his players, and though he bemoaned the way in which Naismith's goal was conceded, he admitted his side could do nothing about Mirallas' stunning winner.
"I thought for the first 70-odd minutes were comfortable in the game, and never looked under threat," he said.
"We looked comfortable but the first goal was really poor from our point of view. But with the second goal you have to hold your hands up and say it was a world-class free kick.
"I don't think we deserved to lose the game."