It had looked as if the Gunners were heading out at the third-round stage for the first time since 1996 after Robert Snodgrass' penalty early in the second half.
However, with just moments remaining, substitute Theo Walcott was tugged back by Ben Parker, and Fabregas stepped up to blast the spot-kick past Kasper Schmeichel, who had been in superb form.
With a two-legged Carling Cup semi-final ahead, and also fighting for the Barclays Premier League ahead of the resumption of European action against Barcelona, Arsene Wenger will not have wanted to add another fixture into their congested schedule.
However, after again failing to make the most of first-half chances, notably through Andrey Arshavin, it could have been a much worse outcome.
Despite making nine changes from the side which drew against Manchester City on Wednesday night, there was no sense of this being an under-strength Arsenal XI as it included the likes of 10-goal Marouane Chamakh, Nicklas Bendtner and Arshavin as well as midfielder anchor man Alex Song.
Leeds - who had won at Manchester United in the third round last season - started brightly with Snodgrass dangerous down the left.
Arsenal should, however, have been ahead on 11 minutes when captain Tomas Rosicky's pass split the defence to release Arshavin.
The Russian was clean through, but Kasper Schmeichel stood up well to make an impressive block at the edge of the area.
At the other end, Wojciech Szczesny was out quickly to save at the feet of Leeds striker Luciano Becchio.
Schmeichel, with his dad Peter watching in the stands, was in action again to beat away Arshavin's angled drive.
Arsenal continued to press and Leeds captain Jonathan Howson was alert to hack a flick-on from Sebastien Squillaci off the line following a corner, with Denilson's 25-yard effort then pushed behind by the busy Leeds keeper.
Alex Bruce made a fine saving tackle as Bendtner looked to skip into the Leeds area, before Chamakh headed straight at the keeper from a corner.
The visitors were defending stoutly, but again had Schmeichel to thank when he saved at the near post from Bendtner.
Arsenal had plenty of pressure at the start of the second half, but were behind from the penalty spot on 54 minutes.
There could be little complaint as Denilson upended Max Gradel and Snodgrass stepped up to beat Szczesny with a low strike into the right-hand corner and send some 9,000 travelling Leeds fans absolutely crazy.
It was no surprise to see captain Cesc Fabregas introduced from the bench with 30 minutes left, replacing Song.
However, Arsenal were indebted to Szczesny for keeping it at only 1-0 when the young Pole produced a brilliant save to deny Becchio's point-blank header.
Wenger introduced England winger Theo Walcott, replacing Chamakh, with just over 20 minutes left to save the Gunners' FA Cup dream.
Snodgrass bent a 25-yard free-kick just wide as Leeds looked to kill off the tie.
Arsenal aimed to get the ball out to Walcott at every opportunity, but were facing a wall of determined blue shirts in the final third.
Wenger's last throw of the dice came after 79 minutes when Rosicky was replaced by Carlos Vela, the Mexican forward who is set to go out on loan.
Walcott was through with three minutes left, but his volley was plucked out of the air by Schmeichel and then felt he had a penalty after a challenge by Bruce, only for the referee to give a goal kick on advice from his assistant.
Phil Dowd, however, did point to the spot moments later when Walcott was tugged back by Ben Parker - and Fabregas stepped up to fire the penalty past Schmeichel.
Bendtner drilled a decent chance wide after a long ball up field, before Schmeichel produced a brilliant one-handed save to deny Denilson to force a replay at Elland Road.