Goals from Alexis Sanchez and Aaron Ramsey earned Arsenal a shock but fully deserved victory at Wembley.
But amid the jubilation at ending the season with a trophy, Wenger was unable to disguise his anger with the Arsenal fans.
Before the match, Wenger had described the criticism he received as a "disgraceful".
And after the final whistle, the Frenchman once again made clear his displeasure at the behaviour of some Arsenal fans.
Arsene Wenger lifts FA Cup as Arsenal players celebrate Sat, May 27, 2017
Arsene Wenger was the centre of attention during the wild Arsenal celebrations
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Arsenal celebrate winning the FA Cup against Chelsea
Some of what has been going on during games this season has been unacceptable in my view
“I am in a public job, I know I have to face criticism and I accept that,” he said.
"I don’t mind facing criticism from supporters or from journalists, I will always try to answer it if I can, but once the game starts you should stand behind your team.
"Some of what has been going on during games this season has been unacceptable in my view, if you are a fan you should support your team.
"We have played some games in quite a hostile environment and I don’t think that has been helpful."
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Arsenal boss Arsene Wenger holds the FA Cup
Wenger is scheduled to meet with the Arsenal board, including Stan Kroneke, next week for talks over his future.
But when asked if the FA Cup success was the perfect time to leave, Wenger remained coy.
"There is no perfect time to depart, I just want to do what is best for the club,” he said.
"It is not about popularity it is about competence. You can see from today that we are a good side that can get better still with a little strengthening, I spend every minute of my season with commitment to this club. I refused to give up hope and so did my team.
"That’s what makes me so proud, and if I am still here next season I will have the same attitude."