Arsene Wenger praises Gunners' team spirit
Arsenal were seeking their first trophy since 2005 but looked set to suffer more disappointment after finding themselves 2-0 down after just eight minutes at Wembley following goals from James Chester and Curtis Davies.
Wenger told ITV: "We wanted to make history tonight and win the game, and we made history in both ways - how not to start a cup final and how to come back.
"I'm very happy for the players because I think it was a turning point in the lifetime of this team, and to know they can win in that way was very important today."
Wenger admitted it would have been "terrible" had they not been able to fight back, but said: "We came back. I've praised many times the spirit of this team and I'm very proud of that today.
"Congratulations to Hull, they played fantastic for the whole game. We won it in the end but they played very well."
Ramsey said it was a dream come true for him to score the winning goal and hopes this is just the start for Arsenal as they look for more honours.
"We showed great strength and character today to come back to win it. To score the winner in extra-time is something I've dreamed of for many, many years," he said.
"It's definitely up there with the most important goals I've every scored. Hopefully this will be the first of many trophies in years to come.
He added of finally ending Arsenal's nine-year wait for a trophy: "The manager and players have been under a lot of pressure over the last couple of years, and this one's for the manager, for the fans, who've always believed in us as well even when things weren't going as well."
"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 1.
"I couldn't ask any more of them, but today it wasn't to be for us, which is unfortunate.
"The vast majority (of the players) 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."