In a spiteful game that saw Arsenal's Nicklas Bendtner and Everton's Mikel Arteta sent off, the Gunners claimed the three points that take them back above Manchester United, shock losers at West Ham today.
"I can understand that they repeated this tactic. This time we won, but it was very close.
"I was tempted to change the team around at half-time, but I felt the reason we were not winning was that we had become nervous after hearing the Manchester United result.
"But we were playing against a good Everton side who stopped us playing, they did well and it was up to us to get the ball down and be more positive. And to be less nervous because we lost too much possession in the first half.
"After the break we were maybe fortunate to score straight away, 1-1, and that changed the game. From then on we looked dangerous, Everton dropped physically and we were ahead very quickly."
Eduardo da Silva struck twice to take Arsenal into the lead, his first goals in the Premier League although he now has nine in 10 games for the Gunners.
Wenger believed his players' character had seen them through.
"The turning point was the fact that we played well with long balls, and we held on when we were down to 10 men and they still had 11," he added.
"They then made that mistake and it killed the game.
"It was hard for Everton because the difference was not as big as 4-1. But they also had a player sent off and it became impossible for them.
"But it was a great result for us, we showed we can battle, we have a great spirit and showed that we can cope with all kinds of challenges - that is always a sign of a very good side."