Goals from Jack Colback and Brennan Johnson either side of half time had given Steve Cooper’s side a healthy lead heading into the second leg at the City Ground on Tuesday.
But Berge’s header after 91 minutes ensured the Blades travel to Nottingham with plenty still to play for.
United started well but a sloppy goal against the run of play turned the match in Forest’s favour and the visitors squandered several opportunities to double their lead in the first period.
The home side improved after the break but their chances of turning the tide were dealt a big blow following a clinical counter attack before Berge’s late effort helped to lift spirits.
Scroll down for our player ratings.
2. Wes Foderingham - 8
First-half hero for the Blades. No chance with either goal. did well to save the initial effort from close range. Made no fewer than five saves in the first half to keep Forest at bay.
Photo: Isaac Parkin
3. Ben Osborn - 6
Can't be blamed for slipping over before Forest's first goal. Battled hard before being replaced by Jebbison.
Photo: Simon Bellis
4. Chris Basham - 7
Twice denied near-certain Forest goals in the first half with some last-ditch defending. Drove his side on with several surging runs too.
Photo: David Klein