"I was not good enough," Swiss Wawrinka, playing his first match since losing to Roger Federer in the Australian Open semi-finals, said. "I started well but it was a tough match. It was his day."
The 24-year-old Dzumhur, making his debut at the Dubai tournament, recovered from losing the first three games to outplay the second seed and three-times grand slam champion.
He will next play Spain's Marcel Granollers next.
Top seed and world number one Andy Murray recovered from losing serve early in his match against Tunisia's Malek Jaziri before posting a 6-4 6-1 victory.
Murray, who suffered a bout of shingles since losing early in the Australian Open, was relatively untroubled as he set up a second round against Spain's Guillermo Garcia-Lopez.
(Writing by Martyn Herman; Editing by Alison Williams)
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