The 24-year-old, ranked second in the WTA standings, lost the opening set despite serving for it at 5-4, before dominating the second and third sets.
The turning point came in a lengthy third game of the second set, where Świątek saved three break points.
The match lasted two hours and 17 minutes.
"It wasn't an easy match. Even in the second and third sets I had to work hard to win rallies against Daria," Świątek said afterwards, admitting she struggled to adapt to the cooler conditions and wind.
Świątek, who won the tournament from 2022 to 2024, will face Greece's Maria Sakkari in the last eight.
She is the top seed following the withdrawal of world number one Aryna Sabalenka.
Poland's Magdalena Fręch and Magda Linette were eliminated in the second round earlier this week.
The Qatar TotalEnergies Open tennis tournament in Doha runs until Friday, 14 February.
(ał)
Source: PAP