Brennan Johnson opened the scoring in the 19th minute with a sliding finish to give Forest a 3-1 lead on aggregate, seemingly putting the tie to bed.
John Fleck equalized with quarter of an hour left, forcing the match to extra time where neither side could find a winner meaning a penalty shootout would determine who would play in the richest game in the world.
Story of the match
Sheffield United started the brighter of the two sides and they had their first chance on ten minutes as from a throw, John Egan was left unmarked in the center of the box, but he guided his header wide.
Philip Zinckernagel started the hosts' next passage of play that ended when a shot by Surridge was blocked by Fleck.
Shortly before the half-time break, Fleck put a body check on Johnson, but referee Michael Oliver let play continue, infuriating the Forest bench.
Two minutes after the restart, Sheffield United found a route back into the tie when McKenna held up Ndiaye, but the ball broke for Sander Berge, who turned it back into danger and Gibbs-White calmly finished.
The visitors applied more pressure looking for the goal that would level the tie, coming closest from a series of corners and throw-ins when Fleck's delivery found Berge, whose header was just off.
On 75 minutes, the goal the Blades were searching for arrived as George Baldock charged down the right and his ball across goal found Fleck, who had the simplest of tap-ins to make it 3-3 on aggregate.
With neither team able to find a winner, the tie headed to extra-time, Forest having the better of the chances. Substitute Joe Lolley slipped with the goal at his mercy and Johnson volleyed straight at Wes Foderingham.
In the 115th minute, Samba produced a brilliant save as he denied Ndiaye's close-range volley, ensuring this pulsating encounter would head to penalties.
In the shootout, Samba saved from Oliver Norwood and Conor Hourihane and Johnson and Cafu put the Reds in immediate control, beating Foderingham. After Berge and Steve Cook exchanged goals, Ndiaye was tasked with keeping the Blades in it and he did so.
Lolley could have sent Forest to the final, but he blazed over. Samba then completed his outstanding night by saving from Gibbs-White to send Forest to Wembley for the first time in 23 years.
Man of the match: Brice Samba
It couldn't be anyone else. His save at the end of added extra time was as important as the three he made in the shootout. He stepped up when Forest needed him most and is the reason the Reds are 90 minutes away from a return to the Premier League.