Wenger's side came from two goals down against Hull City after eight minutes to triumph 3-2 in extra-time at Wembley, clinching Arsenal's first trophy in nine years.
Arsenal will begin their defence of the FA Cup on Sunday when they host Hull at the Emirates.
It has been a frustrating decade for Wenger and Arsenal, with the London club not having won the Premier League since the 2003-04 season and falling behind the likes of Manchester United, Manchester City and Chelsea in the pecking order.
But, when asked if he would have "walked away" if Arsenal had succumbed at Wembley, Wenger said: "Walked where? To walk where? Walk away, no.
"Look I just won my 600th game out of 1000, that means I won 60 per cent of the games and after 18 years it would not be serious to walk away for losing one game.
"I do my job as well as I can and I respect my contract always."
The French manager conceded, however, that Hull's performance just over seven months ago makes him wary of facing Steve Bruce's men again on Sunday.
"I think we were highly focussed on the FA Cup last year and we knew it was the kind of trophy we could win," Wenger said.
"We had a difficult run and we have to be cautious on Sunday because Hull is a difficult team to play.
"It's a bit of a replay of the FA Cup final."
While Wenger is keen to retain the trophy, he also has his sights set on other silverware, with Arsenal sixth in the league and facing his former club Monaco over two legs in the UEFA Champions League last 16.
"We have a big job to do on Sunday and then come back in the Premiership, and then we have the Champions League," he added.
"What is for sure is that we have room for improvement and we have to manage to do that quickly."
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