The 67-year-old also became the FA Cup's most successful manager as he claimed a seventh winners' medal.
The 2-1 victory over London rivals and newly-crowned Premier League champions Chelsea came courtesy of a controversial Alexis Sanchez opener on five minutes, with Aaron Ramsey then heading Arsenal back ahead three minutes after Diego Costa's deflected equaliser.
Sanchez appeared to handle the ball in the build-up to his strike, which was provisionally ruled offside until referee Anthony Taylor over-ruled his assistant.
Chelsea got themselves back into the contest as Costa struck despite the fact Victor Moses had been sent off having been shown a second yellow card following a dive inside the Arsenal box.
After finishing fifth, Wenger and Arsenal missed out on Champions League football for the first time in 19 years.
That came on the back of a 10-2 aggregate defeat to Bayern Munich in this season's competition, with some sections of supporters calling on Wenger to leave this summer, rather than sign fresh terms when his current contract expires next month.
The Frenchman has yet to announce publicly whether he will be manager at the Emirates Stadium next season but did admit his future would become clear following a pre-arranged meeting.
"We have a board meeting on Tuesday, and on Wednesday or Thursday it will all be very clear," he said.
Asked if delivering the FA Cup would be a perfect way to say goodbye having been at the club nearly 21 years, Wenger replied: "There is no perfect way.
"I just want to do well for this club and apart from that after that it is down to the board members am I the right man to take this club further and for me to decide if I'm the right man, that is it.
"It is not about popularity, it is about confidence
It would be a bit ridiculous that 20 years depends on one games and the future of the club depends on one games, we will see more next week."
Ahead of the final, Wenger revealed he has never kept a winners' medal in his managerial career.
But will be holding on to this one having moved clear as the most successful boss in the storied history of the FA Cup, eclipsing George Ramsay's six wins with Aston Villa between 1887 and 1920.
"For once I have kept my medal so that means it is a special night for me," he said.
"I'm very proud, if nobody has done it since 18-god knows-when it means it is not easy.
"I'm especially proud of the way we won this season, in the semi-final we had to play against Man City here and in the final Chelsea here, you cannot get tougher and we had two outstanding performances, this cup was won with special spirit and special attitude."
Wenger was once again critical of those Arsenal fans who have demonstrating against him during matches this season.
Planes carrying anti-Wenger slogans, boycotts of home games and signs and banners being displayed inside stadiums came during a blip for Arsenal which Wenger believes cost them dear.
"I'm in a public job, I accept to be criticised and people don't agree with me," he said.
"But once the game starts and you are a fan, you stand behind the team, that is what I have not accepted this season and I will never accept it.
"Outside of the game I never had any word with any journalist in 20 years when they criticised my decisions and it is the same with the fans.
"But when a game starts you support your team
We have played some games in a hostile environment and I never accept that."
Chelsea boss Antonio Conte could not secure a double in his first season in charge at Stamford Bridge and singled out Ramsey's goal and the dismissal of Moses as the turning points.
"I think our start for sure wasn't good enough," he said.
"Arsenal started the game better than us, honestly I think the goal after three minutes in this way, I think it was not disallowed because there is a clear handball but our start wasn't good.
"Then we started to play, in the second half we started very well and created the chances to equalise.
"The second yellow card for Moses was for sure one of the two key moments.
"But to play for 10 men, we were able to equalise and then to concede a second goal was a key moment for the loss."
The former Juventus coach also refuted claims his players struggled to raise themselves for the final having already secured the Premier League title.
"It is not easy but I think that we won the league and it was right to celebrate and enjoy this win," he added.
"I think that my players put all of themselves into the game, maybe our start wasn't good and the impact and approach wasn't good and Arsenal pushed a lot
We paid for this start."
"Per Mertesacker for me is a perfect example for any young professional football player.
"What he did today was just a consequence of an unbelievable attitude every day, even if not selected, until the Everton game in the squad.
"He played his first game tonight and when he was not selected he worked even harder, which is why I pay a special tribute for him tonight."
Source : PA