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.
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13. George Baldock - 6
Replaced Norwood and showed plenty of endeavour to help push his side on. Expect to see him in the second leg. Photo: Simon Bellis
14. Daniel Jebbison - N/A
A late tactical roll of the dice saw him replace Osborn. Photo: Simon Bellis
15. Rhys Norrington-Davies - N/A
A late change for Enda Stevens. Could well replace him for the second leg. Photo: Andrew Yates