Dzumhur, the first man from Bosnia and Herzegovina to play singles in a Grand Slam, was cheered on by a vocal group of supporters from the Balkans region who now live in Melbourne.
Berdych made it to the quarterfinals last year, losing to eventual champion Novak Djokovic.
The No.7 Berdych is again in Djokovic's half of the draw, but the two cannot meet until the semifinals.
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