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Berbatov coolly slotted past Simon Mignolet for his 10th goal of the season.
Fulham claimed a two-goal advantage after 33 minutes when Sascha Riether stormed forward to turn home Dejagah's cross on the counter.
Sunderland were handed a lifeline just two minutes later when Halsey pointed to the spot for a second time.
Phillipe Senderos looked bemused after being penalised for a push in the area, with Garnder making no mistake from the spot to reduce the deficit.