Ben Brereton Diaz sent the already-relegated hosts ahead in the 17th minute from the penalty spot, but Callum Hudson-Odoi netted either side of an effort from Ryan Yates to send Forest into a 3-1 lead.
Burnley suffered a 4-1 defeat at home to Newcastle United on the same afternoon, meaning that Forest are now three points ahead of 18th-placed Luton Town in the table with just two games left.
The Reds have a substantially better goal difference than Luton, so they almost certainly need just one win from their final two league games against Chelsea and Burnley to guarantee Premier League survival.
As for Sheffield United, it is a 26th league loss of what has been a very tough campaign, with Chris Wilder's side now on 16 points from 36 matches, and they have become the first side to concede 100 goals in a 38-match Premier League season with two games still to play.
Forest now three points ahead of the relegation zone
Chris Wood was guilty of missing chances in Forest's clash with Manchester City last time out, and he passed up another opportunity in the early stages of this match, slotting a dangerous Morgan Gibbs-White cross wide of the target.
Forest were threatening, but it was Sheffield United that made the breakthrough from the penalty spot in the 17th minute of the contest through Brereton Diaz.
Gonzalo Montiel was penalised for bringing Brereton Diaz down inside the box, and the Chile international got back to his feet before firing past Forest goalkeeper Matz Sels to make it six Premier League goals for the Blades since arriving on loan from Villarreal in January.
Forest's response was strong, though, and the scores were levelled in the 27th minute through Hudson-Odoi, who expertly picked out the far corner with a brilliant finish after creating a yard of space inside the box.
Callum Hudson-Odoi comes up with a brace for Forest
Instead of Forest pushing on, it was Sheffield United that ended the first half on the front foot, with Selz needing to deny Gustavo Hamer, before Anel Ahmedhodzic fired the ball over the crossbar.
The visitors managed to take the lead in the 51st minute, though, with Yates striking into the back of the net from inside the box after a knock down from Willy Boly.
Hudson-Odoi curled one just wide of the post in the period that followed, with Forest looking for a third, and it arrived in the 65th minute through the former Chelsea attacker, who picked out the bottom corner with a powerful finish.
There was huge pressure on Forest when Sheffield United took the lead, and it was a very impressive response from Nuno Espirito Santo's side, who also sit five points ahead of Burnley in England's top flight.
Sheffield United's afternoon went from bad to worse before the final whistle, though, with Ahmedhodzic sent off after picking up a second yellow card for an elbow on Nicolas Dominguez.