World number 99 Dzumhur broke the five-times champion in his first two service games and raced out to a 4-0 lead before many in the crowd had found their seats at Arthur Ashe Stadium, which had its retractable roof closed due to rain.
Despite some shaky serving, the 27-year-old Dzumhur took the first set thanks to 17 unforced errors from the Swiss.
The tide shifted in the first game of the second set, however, when Federer saved a break point at 30-40 following a wild rally during which both players race from corner to corner before Dzumhur sent a forehand into the net.
From there Federer’s superior serving carried the day as he finished with 16 aces to the Sarajevo native’s two and thumped a rocket that Dzumhur could not return on match point to clinch victory.
Federer, however, has looked vulnerable at the tournament.
He dropped the opening set of his first-round match against qualifier Sumit Nagal and while he may get away with slow starts in the early rounds, he will need to be more consistent if he hopes to win a record-extending 21st Grand Slam title.
Federer will look to get off to a better start when he faces either Lucas Pouille or Dan Evans for a place in the round of 16.
Japan's Kei Nishikori fended off Bradley Klahn 6-2 4-6 6-3 7-5 on Wednesday as the 2014 Flushing Meadows runner-up continued his bid for a first Grand Slam title.
On the women’s side, Madison Keys eased through to the US Open third round, recovering from a slow start and an early loss of serve to beat China's Zhu Lin 6-4 6-1.
Fifth seed Elina Svitolina knocked downed Venus Williams to reach the third round despite an heroic defensive effort from the 39-year-old former champion.
