Arsene Wenger's men came into Saturday's clash as heavy favourites to end their infamous barren run, but found themselves two goals down inside the opening eight minutes.
James Chester diverted the ball beyond goalkeeper Lukasz Fabianski from a Tom Huddlestone strike to send fans in the Hull end of the ground into raptures four minutes in.
The outlook began to look even bleaker for Arsenal when Curtis Davies powered home from a tight angle.
A sensational Santi Cazorla free-kick 10 minutes later gave the 10-time winners hope, before Laurent Koscielny's 71st-minute leveller forced the game into extra time.
Arsenal's comeback was completed in the 19th minute of extra time when Ramsey - the man whose absence through injury so hurt his side's Premier League hopes - prodded home with a first-time strike from a sublime Olivier Giroud back-heel.
Fabianski kept his regular cup berth between the posts, despite doubts over his future with the Emirates Stadium club, while Chester recovered from a hamstring injury in time to step into a new-look Hull back line.
Chester's inclusion proved to be an inspired decision from manager Steve Bruce as the defender stunned Arsenal fans into silence by breaking the deadlock in the fourth minute.
A low 20-yard strike by Huddlestone fizzed towards Chester's feet, and his touch in the centre of the penalty area was enough to send the ball beyond an outstretched Fabianski, finding its way in off the left-hand upright.
In the absence of cup-tied forward pairing Shane Long and Nikica Jelavic, it was Hull's defenders running riot in the opening stages.
Fabianski leapt to his left to keep out an Alex Bruce header, before fellow centre-back Davies rifled home from a tight angle, though question marks were raised over the positioning of the free-kick that led to the goal.
Another Alex Bruce header from a corner threatened to make it a defensive hat-trick for the underdogs, but Kieran Gibbs came to Arsenal's rescue on the goalline.
There was no let-up in the action, and Arsenal were right back in the tie in the 18th minute.
Cazorla curled a 25-yard free-kick into the top right-hand corner of the net - Hull keeper Allan McGregor wrong-footed by the Spaniard's stunning strike.
There was a nervous edge to the match after the break, and Giroud felt he should have been awarded a penalty in the 58th minute after having his shoulder held by Huddlestone.
Further penalty shouts followed as Jake Livermore appeared to handle from a Yaya Sanogo header and Cazorla was felled by Davies, but referee Lee Probert was unmoved by the appeals.
After looking the more likely to find the net in the second period, Arsenal finally pulled level, Koscielny turning the ball in from close range in the aftermath of a contentious corner.
An unmarked Gibbs should have sealed the trophy for Arsenal in the 79th minute, but the full-back blazed over from close range when it appeared easier to score, and McGregor got down well to deny Giroud from 18 yards three minutes later.
Substitute Sanogo sent a strike whistling wide of the left-hand post before the end of 90 minutes, and Giroud rattled the crossbar in the opening stages of the additional period.
The match looked to be heading for penalties as the tempo of the game dropped, but Ramsey popped up to end nine years of hurt for Arsenal and Wenger, though Fabianski's late charge off the line had many a heart in mouth as Sone Aluko's effort drifted agonisingly across the face of goal.
