The minister for higher education, 29, came to the country at the age of five in 1992 as a refugee from Bosnia with her parents fleeing the war in the Balkans.
Hadzialic on Saturday drank two glasses of wine before being stopped on the bridge linking Denmark and Sweden.
She faces a possible term of up to six months in prison after police detected a blood alcohol level of 0.2 gm per litre, Xinhua news agency reported.
"That was the biggest mistake of my life... I will take responsibility. I announce my intention to resign from my ministerial post," Hadzialic said.
"I understand that a lot of people are disappointed. And I am angry with myself, and certainly I deeply regret it," she added.
Explaining her error, Hadzialic said she had drunk one glass of sparkling wine and one of red wine after a night out in Copenhagen.
Four hours later, she set off for Sweden's southern city of Malmo, believing that would have been enough time for her body to get rid of the alcohol.
In 2014, at 27, she became the youngest ever cabinet minister in Sweden's history.
Sweden is one of several European countries which have low alcohol limits for drivers.
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