Hadzialic, 29, is minister for upper secondary school, adult education and training. She was caught by police on Thursday night when she was detected a blood-alcohol level of 0.2 grams per litre, which is just above the legal limit in Sweden, Xinhua reported.
Local media reported she would likely face a fine and a driving ban.
At a press conference on Saturday, Hadzialic said the incident was her "life's biggest mistake".
"I understand that many people are disappointed with me and I am angry with myself and above all I am deeply regretful," she said.
Hadzialic explained that she had been to a concert in Copenhagen on Thursday night with a female friend and drank two glasses of wine early in the evening before setting off to cross the Oresund bridge over to Malmo, southern Sweden, four hours later.
She apparently believed the alcohol would have left her system by then.
However, the Social Democrat minister was then stopped at random in a police alcohol control on the Swedish side of the border.
Speaking to local media, Prime Minister Stefan Lofven said he had a long conversation with Hadzialic before she announced her resignation on Saturday.
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