But first defender Jagielka scored a sensational 35-yard screamer in the 91st minute to leave extra-time beckoning, only for Ndlovu to pounce with the last kick of the game.
The goal and referee Jeff Winter's final whistle seconds later led to a few hundred jubilant Blades supporters storming onto the pitch - and ugly scenes followed.
In response, the Leeds fans started ripping up seats from the top tier of the John Street Stand and throwing them onto the pitch, forcing mounted police to intervene.
They drove the Blades followers back into the remaining three sides of the stadium and beyond the halfway line, and a Football Association inquiry is likely to follow.
The 2,000 fans from West Yorkshire were then kept behind by police as the rest of the ground cleared, and as they waited there were chants of both 'Venables out' and 'Ridsdale out'.
Just two points in the previous six Premiership matches had ensured the wolves were snarling at Venables' door ahead of this clash, with the Leeds fans united on Sunday in their condemnation of the former England manager.
The 1-0 defeat at home to Everton had proved too much for many irate supporters as they bombarded a national radio phone-in that night calling for Venables to either be sacked or quit.
Venables, though, is far too wily and experienced a campaigner to let such abuse get to him and yesterday he made it plain he has no fears over his future after just four months in the job.
Opposite number Neil Warnock, realizing Leeds were in nothing more than a slump as they come to terms with their new manager's demands, but clearly his methods are not working.
Leeds may have controlled the first half in terms of possession, but they lacked a clinical edge to their game, with keeper Paddy Kenny only troubled once other than picking the ball out of the net from Yates' misfortune.
Venables gave Jason Wilcox a shock recall as the winger only returned to training last week due to a back injury having prevented him from playing any part to date in this season.
For Wilcox, it was his first taste of action since the final match of last season and his first start since January 30, and he was at least on hand in setting up the opener.
The 31-year-old played a delightful one-two on the left-hand edge of the area with Harry Kewell, up front with Mark Viduka in the absence of the ankle-injured Alan Smith, before turning in a low cross along the six-yard box.
The 32-year-old Yates, under no pressure with barely a blue shirt within five yards of him, attempted to clear the danger, but only succeeded in inadvertently slicing the ball beyond Kenny.
Leeds had earlier had a strong penalty appeal turned aside by Winter when Stuart McCall appeared to bring down Wilcox, only for the Stockton official to turn a deaf ear.
While the Blades enjoyed nothing more in the opening 45 minutes than two long-range drives from Michael Brown and McCall which barely troubled Robinson, Kenny was called upon to parry at full stretch a Viduka curler, while the Australia international was also inches away with a 22-yard free-kick.
The second half proved a different story for Warnock's side who threw everything into grabbing the equalizer, and at times the Leeds defence creaked and groaned before finally crumbled in the dying moments.
Michael Tonge, likely to be a £3million target for Liverpool when the transfer window reopens in January, showed just why he is coveted by Gerard Houllier as he was sensational.
When Teddy Lucic fouled Tonge two yards outside the area in the 50th minute, it was the 19-year-old who picked himself up to curl a right-foot curler beyond Robinson's left-hand post.
It was Tonge who then capitalised when Viduka shockingly gave the ball away midway inside the Blades' half soon after, the winger embarking on a 50-yard run towards the Leeds area before giving Wayne Allison a clear sight of goal, but his 15-yard shot was wasted as it was straight at Robinson.
The future England international goalkeeper was then left breathing another sigh of relief soon after when Brown was left free inside the area, but again the finish was poor and directed at the 23-year-old.
But Robinson could do nothing about the equaliser as Ian Harte's headed clearance fell to Jagielka and after one touch to control, he then sent a bullet into the bottom right-hand corner.
The Sheffield United fans went into a frenzy, but there was jubilation beyond belief for with the four minutes of injury time elapsed, Ndlovu stabbed home his own rebounded shot which had cannoned off Allison to send the Blades into the last 16 for the first time since 1991.
But the game, not for the first time with Leeds fans, was darkly overshadowed by the sad scenes which unfolded at the end.
Man of the Match: The whole team deserves the award but I will again give it to two players, Michael Tonge and Paddy Kenny.