The Poland international goalkeeper was an FA Cup ever-present for Arsene Wenger's side this season, and he was preferred to first-choice Wojciech Szczesny for the 3-2 victory over Hull in the final.
Fabianski has failed to establish himself as a first-team regular since being signed from Legia Warsaw in 2007, and made only one Premier League appearance this season.
The 29-year-old will be out of contract in the summer having rejected the offer of an extended deal, but he enjoyed his time in the spotlight at Wembley in his 78th appearance for Arsenal.
"It feels great, it feels amazing," Fabianski said. "I don't know how best to express my emotions but it feels amazing.
"It was probably my last game for Arsenal. I don't know what I can say, but if
I end my Arsenal career in that way then it is the best way to finish.
"It is a massive relief because it has been a long time without a trophy, especially after that kind of game when we were 2-0 down in the cup final.
"No-one was expecting us to be two goals down after 10 minutes and to produce that great comeback I think was a massive, massive relief.
"Hopefully the club can now kick on. I think this gives us or gives Arsenal the idea that those players can win trophies, that they are able to."
Meanwhile, striker Lukas Podolski paid tribute to the spirit in the Arsenal camp after they came from 2-0 down to clinch the trophy after an extra-time winner from Aaron Ramsey.
"You win nothing for nine years so it is special for us," the Germany forward said. "We have to come and celebrate now.
"For the fans, for the club, it is something special. You wait and wait for nine years and some people repeatedly said 'Arsenal win nothing, Arsenal win nothing'.
"It is a lot of pressure but in the end it is a cup game and, in the end, it can happen. We were 2-0 down and came back, it was a great match and we had the cup in our hands.
"We have a great spirit in the team. Everybody is a part of this and you see in the last matches we did well, getting fourth place and now winning the FA Cup.
"It was a special cup game because Hull fought great and played great but in the end we have it."