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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5. John Egan - 7
Generally solid, but was robbed for the second goal. Saw a headed effort cleared off the line from a corner and made one or two timely interventions to keep Forest at bay. Produced another goal-saving sliding interception in the second period. Photo: Simon Bellis
6. Jack Robinson - 6
Appeared to have a hand in United's goal. Generally did OK. Photo: David Klein
7. Enda Stevens - 5
Given a tough time by Brennan Johnson and Djed Spence. Replaced by Rhys Norrington-Davies ahead of what could be a like-for-like change in the second leg. Photo: Darren Staples
8. John Fleck - 6
Perhaps lucky to have not seen red for a diving lunge on Zinckernagel. Like Norwood, struggled to exert control, which was central to United's struggles. Photo: Simon Bellis