The Gunners were well in control after Alexis Sanchez's opener, but failed to press home their advantage and the Toffees, low on confidence after just one win in 10 league matches, found a spark to ignite their game.
Seamus Coleman scored an important equaliser just before the break to cap a lengthy spell of growing pressure and four minutes from the end Ashley Williams headed in the winner before Phil Jagielka was sent off for a second bookable offence in added time.
It marked an significant turnaround for Williams himself, scoring his first goal since arriving at Goodison in the summer, having clattered team-mate Idrissa Gana Gueye with a knee-high challenge in the build-up to Arsenal's goal, which deflected in off him.
"That is important
Every player needs to show that personality," said Koeman.
"Players like Ashley, with that experience, if they show the feeling and confidence younger players will pick up that confidence.
"It is good to score the 2-1 because that will give the players the boost they need."
Midfielder Ross Barkley was recalled to the starting line-up after two matches on the bench and also seemed to have got the message his axing was meant to deliver.
"He was one of the players who start nervously, but everybody knows Ross can be a fantastic player," added Koeman.
"After 20 minutes he was strong, aggressive, kept the ball in midfield
In the Premier League you need to be aggressive."
Arsenal manager Arsene Wenger was unhappy with the way his side failed to take advantage of their early goal - missing the opportunity to go top of the table, but was even less impressed with referee Mark Clattenburg for awarding the corner which led to Williams' winner.
"Of course, it was no corner
You can see it was no corner," he said after the team's 14-match unbeaten league record was ended.
"I am really disappointed in Mr Clattenburg
He is in a really good position to see it and it is not the first time we are really unlucky with his decisions.
"The referee made his decision, but that does not explain why we did not head the ball.
"We have to look at ourselves
We started well and after that lost a bit of urgency.
"We were a bit too comfortable and Everton made it very physical.
"I believe they disturbed our game and we created less
In the end unfortunately we didn't take our chance or two.
"They played a cup game
That is a big difference from what I saw recently from them.
"I don't think they can repeat that in every game."
Source : PA