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The 20-year-old, who dumped out Roger Federer en route to the Australian Open semi-finals last month, saved two match points late in the third set but still slumped to a 6-4, 1-6, 7-5 loss.
It was Dzumhur's first win since October after a run of four straight defeats, and Tsitsipas was left frustrated by an erratic display.
"I lost to myself, not to him," said the world number 12.
"He fools you, waits for you to make mistakes. He stands eight metres behind the baseline and plays with your strengths. He played nothing crazy or special, but you don't get any rhythm with him.
"The momentum was on my side but he kept putting the ball into the court and running."
Tsitsipas was seeded third at the Ahoy Arena, but has now lost three of his last four matches after following his Melbourne last-four thrashing by Rafael Nadal with a quarter-final exit at the hands of Gael Monfils in Sofia last week.
Stan Wawrinka, the 2015 champion, ended a three-match losing run against Milos Raonic as he defeated the fourth-seeded Canadian 6-4, 7-6 (7/4) to reach the last eight.
Wawrinka, ranked 68th after spending last season recovering and rehabbing from two 2017 knee operations, is fast gaining momentum in 2019.
The 33-year-old Swiss saved the only break point he faced while holding big server Raonic to a relatively modest 15 aces.
"I was moving much better in this match than some of the others against him," said the three-time Grand Slam title winner.
"This was my best match in many months. Some of my losses this season were close calls. I'm trying to push myself, I'm looking for results."
Wawrinka will play his second quarter-final of the season after Doha when he takes on Denis Shapovalov after the Canadian teenager beat former champion Tomas Berdych 6-4, 6-3,
Earlier on Wednesday, Jo-Wilfried Tsonga continued his strong form with a 6-1, 6-4 defeat of Italy's Thomas Fabbiano.
The Frenchman, who missed seven months of last season after knee surgery, continued his momentum after winning last weekend's tournament in Montpellier.
Tsonga, ranked 140th, spent just 66 minutes in dispatching qualifier Fabbiano.
The former Australian Open finalist, 33, was pleased with his showing as he won his 10th match from 12 since the start of the year.
"I played well in the first set and kept my level in the second," he said. "My opponent raised (his) in the second but I held on."
Tsonga next plays Dutch wildcard Tallon Griekspoor, who knocked out second seed Karen Khachanov on Tuesday and admitted he does not know much about the 211th-ranked home hope.
"I really don't know about him, but we will work on it," Tsonga said.
Russian fifth seed Daniil Medvedev won his opening match of the week, beating France's Jeremy Chardy 7-6 (7/2), 6-2. Hungarian Marton Fucsovics put out 2016 Rotterdam champion Martin Klizan 7-6 (7/4), 6-4.
Like Tsonga, Medvedev, who turned 23 on Monday, also arrived in the Netherlands after winning a title, having claimed his fourth ATP tournament win in Sofia.
Monfils reached the quarter-finals on Wednesday at the expense of Italy's Andreas Seppi, 4-6, 6-1, 6-3.
Results from the ATP tournament in Rotterdam on Wednesday (x indicates seeding):
Second round
Stan Wawrinka (SUI) bt Milos Raonic (CAN x4) 6-4, 7-6 (7/4)
Denis Shapovalov (CAN x10) bt Tomas Berdych (CZE) 6-4, 6-3
Gael Monfils (FRA) bt Andreas Seppi (ITA) 4-6, 6-1, 6-3
First round
Marton Fucsovics (HUN) bt Martin Klizan (SVK) 7-6 (7/4), 6-4
Damir Dzumhur (BIH) bt Stefanos Tsitsipas (GRE x3) 6-4, 1-6, 7-5
Daniil Medvedev (RUS x5) bt Jeremy Chardy (FRA) 7-6 (7/2), 6-2
Jo-Wilfried Tsonga (FRA) bt Thomas Fabbiano (ITA) 6-1, 6-4
