Mesut Ozil had a shot back-heeled off the line by Gary Cahill while Chelsea had an incredible escape when Danny Welbeck's header thumped against the post and Ramsey bundled the ball back against the same upright.
Bellerin shot just wide and Ozil hit the post after cutting inside Cesar Azpilicueta, but just as in the 2014 final against Hull City, Ramsey's goal was to prove the victor.
A larger than usual security presence welcomed supporters at a time of heightened alert following Monday's terror attack.
"At the last minute they chose to wear the jackets and the black armbands were left where they were".
Alex Sanchez opened the scoring early for the Gunners, as Arsenal took a controversial 1-0 lead within five minutes.
The winning goal was scored by Aaron Ramsey.
Amid confusion, referee Anthony Taylor consulted his assistant and the goal was allowed to stand.
The Blues did have a half-chance on 15 minutes but Diego Costa's snatched shot just inside the area was block.
Mesut Ozil then really should have scored after latching onto a Sanchez pass before beating Courtois with his scuffed shot that Gary Cahill sensationally back-heeled off the line.
Costa was thwarted by captain Per Mertesacker, starting his first match in 392 days, and Ospina as Chelsea looked to level, but their defence was being too easily exposed.
Wenger became the competition's most successful manager with a seventh title in his 21-year reign, triumphs that have papered over some of the cracks as Arsenal has repeatedly come up short in the Premier League and Champions League. Moses darted into the right side of the Arsenal area and appeared to dive, with the referee subsequently dismissing the Nigeria global for a second yellow card.
Many football fans were simply happy to see Chelsea finally lose after a storied season.
And then there was the perplexing choice of David Ospina, his manager deciding his loyalty should go towards a second-choice goalkeeper who will leave this summer while leaving out the one, Petr Cech, who will stay.
Chelsea's reprieve was short-lived, though.
It was an incredible response that Bellerin almost added to after a stunning solo run and, having seen Ospina stop Costa grabbing his second, Ozil sent an effort off the base of the post. The linesman flagged for offside but the Arsenal players protested.