Goalkeeper Brice Samba was Forest’s hero, saving three Blades spot-kicks in the shoot-out to earn Forest their first ever trip to the EFL Championship play-off final.
Steve Cooper’s men will face Huddersfield Town in the final on May 29 with a place in next season’s Premier League for the winners.
Forest, leading 2-1 from the first leg, looked in charge when Brennan Johnson put them ahead on the night and 3-1 up on aggregate.
But United fought back and goals from Morgan Gibbs-White and John Fleck made it 2-1 to the Blades on the night and 3-3 on aggregate after 90 minutes.
Extra-time brought no further goals and the tie went to penalties and it was Samba who stepped up for Forest with three superb saves to send the Reds to Wembley.
Forest boss Cooper, speaking to Sky Sports, said: "The most important thing is that we're through.”
"We've been brilliant this season, we gave automatics a real push.
"We were outstanding at the weekend and that's helped us get over the line.
"But these play-offs are a little bit different and you have to do a bit of everything to get through, and that's what we've done.
"The support was incredible.
“We went 2-1 down and the crowd got better and louder.
"This is a magical place anyway, but it was particularly good tonight and I'm very grateful for them getting behind us and sticking with us.
The night was sadly marred at the end of the game when Sheffield United striker Billy Sharp was assaulted by a fan as the crowd spilled on to the pitch at the final whistle.
Police have now arrested a man in connection with the incident and are continuing to liaise with both clubs as enquiries continue.
Chief Inspector Paul Hennessy, of Nottinghamshire Police, said: “A 31-year-old man has been arrested on suspicion of assault in connection with this incident.
"He remains in custody and will be questioned by officers.”