First-half goals from Danny Welbeck and teenager Alex Iwobi were enough for a first Premier League victory in a month which came as a welcome relief after exits from the FA Cup and Champions League in the same week.
The Gunners remain some distance from leaders Leicester but at least they retain an interest in the title race going into April.
"The character of my players was questioned
We know if we don't win today we were out of it
I think it was a mental test for the side," said Wenger.
"I am not masochistic enough to consider us out
Mathematically we have a chance and a game in hand at home.
"We are not the flavour of the week but if we don't believe, no one will do it for us.
"Even if the mind is sceptical, we have to give our best in every single game
We have to be humble and fight in every game like we did today then we never know.
"Of course we need some performances and some good results from the others
We cannot help that."
Wenger was impressed with 19-year-old Iwobi, the nephew of former Premier League star Jay-Jay Okocha, who opened his account for the club with a coolly-taken goal.
"He is an intelligent boy who loves football, is very passionate about the game, keen to learn and very humble as well," added the Gunners boss.
"He is 19 years old and has been at our club since the age of eight.
"I took him in pre-season and I liked what he did
After that he improved very quickly
He did very well
I integrated him and played him."
Defeat for Everton, their eighth at home this season, highlighted the inconsistency of their form which has blighted their campaign.
"Today was a one-off
I don't think we were ourselves in any part of the game and it was frustrating to see such contrasting performances," said manager Roberto Martinez.
"Seven days ago (when they knocked Chelsea out of the FA Cup) we had an intensity which was joyful.
"Today we had doubt and we were pedestrian
We looked like the team that played on Wednesday and travelled in Europe
Arsenal looked the fresh team.
"I don't think we had one period in the game when we looked like ourselves.
"Normally we create opportunities
We never did that
We were very slow in our build-up.
"It looked like a fearful performance."
Source : PA
Source: PA