The Welsh midfielder claimed the winner as Arsenal came from two-goals behind before claiming a 3-2 extra-win over Hull City.
Ramsey's winner came just 11 minutes before penalties would have been needed as he swept the ball home following a neat back-heel from Olivier Giroud.
"It is mind blowing, I dreamt about this day as a kid and it hasn't quite sunk in yet," he told Sky Sports.
"I am more delighted for the team that we have won a trophy this year, we deserved it and the manager deserved it, he has stuck by his players after we were questioned a few times but he never had any doubt and we believed in it ourselves and hopefully this will be the start of many.
"I couldn't do this without my team-mates and I am delighted to be in this team."
Ramsey admits that getting back into the game quickly through Santi Cazorla after going two-behind - was crucial.
"We knew ourselves that Santi Cazorla's free kick was important and we had time to get back into the game but I thought we were in control," he said.
"In extra-time at 2-2, things go through your mind like penalties but it is important to keep going and I managed to get into the box and get that one."