The Blades had dragged themselves back into this semi-final despite losing the first leg 2-1 and trailing at the City Ground to Brennan Johnson's opener as second-half goals from Morgan Gibbs-White and John Fleck took the tie to extra time.
As the tie finished level at 3-3 on aggregate after 120 minutes, Congolese goalkeeper Samba denied Oliver Norwood, Conor Hourihane and Gibbs-White to send Forest through to a showdown against Huddersfield at Wembley Stadium on Sunday May 29.
Johnson, Cafu and Steve Cook all scored their spot kicks and while Joe Lolley fired over it was inconsequential after Samba saved from Gibbs-White to prompt a pitch invasion from jubilant Forest fans.
Forest, bottom of the second tier after seven games when Steve Cooper took charge in September, are now just one win away from a return to the top-flight after a 23-year absence.
The hosts held the upper hand after the first leg although they conceded late on at Bramall Lane and were pushed all the way.
Forest were initially tentative, and while John Egan's bullet header missed the target it was a warning shot.
Samba saved superbly soon after, rushing out of his goal to smother Ndiaye's shot after he was put through from Gibbs-White's exquisite pass.
From nowhere, Forest reclaimed their two-goal cushion as Scott McKenna's long ball was collected on the left by Sam Surridge, whose square pass allowed an outstretched Johnson to send beyond Wes Foderingham on 19 minutes.
But any thoughts the visitors were becoming desperate were extinguished two minutes after halftime to bring the tie to life.
Gibbs-White provided the finish, sweeping across Samba from Sander Berge's low ball to the near post.
Nerves around the ground heightened as United enjoyed a spell of sustained pressure, and they were rewarded in the 75th minute as George Baldock's cross after a lung-bursting run down the right was turned in by Fleck to level the scores on aggregate.
Gibbs-White and Keinan Davis had chances in the first period of extra time and Sheffield United might have won it late on, but Samba produced an astonishing point-blank save to deny Ndiaye from six yards.
In the shootout, Samba guessed right to deny Norwood then tipped Hourihane's effort on to the bar before saving in the corner from Gibbs-White to spark frenzied scenes on a memorable evening for Forest.
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