That came after a remarkable opening to the game which saw Hull race into a two-goal lead within nine minutes.
James Chester side-footed home from a Tom Huddlestone drive before captain Curtis Davies fired into the net from an acute angle after Alex Bruce's header had hit the post.
Arsenal did not have to wait too long to get back into the game as on 17 minutes Santi Cazorla curled home a superb free-kick from 30 yards.
Hull looked like they might just hold out but with 19 minutes left they failed to deal with a corner and Laurent Koscielny turned and levelled from inside the six-yard box.
Then with penalties just 11 minutes away, Arsenal broke Hull hearts as Ramsey fired home low past Allan McGregor after great work from Olivier Giroud.
The win ends Arsenal's own trophy drought as they claimed their first silverware in nine years.
Few would have predicted a flying start for Hull, but that is what arrived as the Tigers swept into the lead on four minutes.
A corner from the right by Stephen Quinn was floated out to Huddlestone - the usual corner taker.
His scuffed shot bounced into the turf through to Chester, who flicked it back into the bottom left corner just past Lukasz Fabianski - again preferred to number one Wojciech Szczesny following his heroics in the semi-final shoot out - as Arsenal had no man on the far post.
Rather than respond, the Gunners suddenly found themselves 2-0 down.
Again it was a delivery into the penalty area which caused problems, this time a ball across from Quinn out on the left.
Bruce's header was deflected onto the post by Fabianski and Davies reacted quickest to smash the ball home from a tight angle.
It was almost 3-0 on 13 minutes when another corner was headed back towards the far corner by Bruce and as the ball looked to be dropping in, this time Kieran Gibbs knocked it off the line.
Then Arsenal got themselves back into the game. On 17 minutes, Cazorla was tripped some 30 yards out by Bruce.
The diminutive Spaniard elected to take the free-kick himself, which he promptly lashed into the top right corner to haul Arsenal back into the final.
To their credit Hull, beaten 3-0 at home by Arsenal last month, regrouped and then offered some threat of their own as the Gunners pushed for an equaliser.
Arsenal continued to ask questions into the second half, with Ozil's clever dragged cross through the six-yard box just too far ahead of Podolski.
With 30 minutes left, Wenger replaced Podolski with Yaya Sanogo as Arsenal went with two up front and they were soon level.
Sanago's flick at the near post was blocked behind for a corner, although replays suggested goalkeeper McGregor did not get a touch.
The ball was swung into a crowded six-yard box where it cannoned off two bodies before dropping to Koscielny, who hooked it over the line before being clattered by the goalkeeper.
Sone Aluko replaced Quinn to give Hull some much-needed impetus in attack.
Gibbs crashed the ball over the bar after it was dropped to him six yards out at the back post following strong work by Sanogo.
Cazorla looked to have been nudged over by David Meyler, but it was no surprise the referee waved play on before McGregor pushed a low angled drive from Giroud behind.
Sanogo lashed a 20-yard effort only inches wide but that was the last chance before extra-time.
Into extra-time and Arsenal almost went ahead straightaway. Ramsey caught Davies in possession to drift over a deep cross, which Giroud looped back onto the bar.
Wenger sent on Jack Wilshere and Tomas Rosicky, replacing Ozil and Cazorla, for the final 15 minutes, and it was Arsenal who emerged victorious following and instinctive finish from Ramsey after a great back-heel from Giroud.