The lawsuit was filed by immigrant-rights advocates seeking to overturn the executive order, The Washington Times reported.
Brewer said Friday she was "not surprised" by the lawsuit.
"The state is the one that licenses the people to be able to drive around the streets. It's not the federal government and we don't license kids under 16, we don't license DUI drivers, and our laws are very clear and I took an oath to uphold that," she said.
The lawsuit, brought forth by the American Civil Liberties Union, the National Immigration Law Center and the Mexican American Legal Defense and Educational Fund, alleges Brewer's order is unconstitutional because it is superseded by federal law, The Arizona Republic reported.
"Not only is this wrong and mean-spirited, it's completely out of line with the constitutional guarantees that prevent government from treating people differently based on race, ethnicity or, in this case, immigration status," said Alessandra Soler, executive director of the ACLU of Arizona.
Gubernatorial spokesman Matthew Benson, however, disagreed, saying, that unlike "all previous classes granted deferred action, the DACA program is neither congressionally authorized nor enshrined in federal law."
"The legal limbo now faced by DACA recipients is not due to any action by the state of Arizona or Arizona voters. Rather, it is due to President Obama's decision to pursue this program via executive action rather than through the proper legislative process."