Arsenal's hopes of ending a nine-year trophy drought appeared slim after centre-backs James Chester and Curtis Davies put Hull 2-0 up after only eight minutes of Saturday's final at a humid Wembley Stadium.
But Santi Cazorla's inch-perfect free-kick halved the deficit and after Laurent Koscielny scored a brave equaliser with 19 minutes remaining, Aaron Ramsey drilled home a 109th-minute winner in extra time.
Having seen his side falter in similar circumstances over recent years -- most notably in 2011, when they lost the League Cup final to Birmingham City -- Wenger was delighted by the spirit his players showed.
"The mental aspect is very important in a game like that when everybody expects you to win and you're 2-0 down," he said.
"Of course you're under pressure. Everybody thinks you have won the game before you started.
"The cup final in England is special. We have seen it last year (when Wigan Athletic beat Manchester City); we have seen it again this year.
"It was an important moment in the life of this team, because to lose today (Saturday) certainly would have been a major setback and to win will be a good platform to build on and come back even stronger next year."
On having finally ended the club's long wait for a trophy, he added: "It's a big, big moment of happiness. The happiness is linked sometimes with the suffering and the time you have to wait."
Wenger has now won five FA Cups -- equalling former Manchester United manager Alex Ferguson's tally -- and he identified Saturday's triumph as the most impressive of them all.
"It was more important today than all the others," he said.
"We've made the Double twice, so you have already won (another trophy) and you're not under the pressure we were under today. I've won now five times. It's not too bad."
- 'Ran out of juice' -
Having pledged to sign a new contract in the coming days, Wenger will in time turn his thoughts to making a more sustained challenge for the Premier League title next season.
Arsenal made a storming start to the league campaign that has just ended, only to finish seven points below champions City, and Wenger warned that next season's title race will be every bit as keenly contested.
"You have to accept that in England the Premier League is very tough," he said.
"You look at the top four this year -- you have no Man United, you have not Everton, you have not Tottenham.
"All these clubs have invested a huge amount of money and the clubs who won the Premier League have invested incredible amounts of money, so it's difficult to beat them, but we'll try."
Hull manager Steve Bruce, whose side were already guaranteed to qualify for next season's Europa League, said that he could not fault his players after the first FA Cup final appearance in the club's 110-year history.
"Unfortunately we ran out of juice. You could see that. All week we've been carrying people and they hurt us at the finish," he told BT Sport.
"I couldn't ask any more of them, but today it wasn't to be for us, which is unfortunate.
"The vast majority are the team that got us promoted last year, so all credit to them. They've given everything, but in the end we just fell short on the day."
Ramsey's goal -- a crisp half-volley from an Olivier Giroud back-heel -- capped a fine individual campaign in which he scored 16 goals and emerged as Arsenal's most influential player.
The Wales midfielder was overlooked in the voting for the Professional Footballers' Association Team of the Year, however, and Wenger queried his exclusion.
Told that Ramsey's goal was a fitting end to his season, Wenger joked: "He was not in the team of the year in England, so don't tell me that. I think he deserved to be in. He popped up at the right moment again today."