However, she criticized the bill's opponents who say it will disenfranchise millions of voters.
"Each of the arguments that have been laid out to criticize this bill are baseless speculation from the radical left because they want illegal aliens to vote in our elections," Noem told reporters during a news conference.
"They want to disenfranchise American citizens by telling them that their votes don't matter. There's only one reason that anyone would oppose this bill, and that's because they would want to cheat.
"They want illegal people and aliens in this country to be able to vote for them and to rob the United States citizens of their vote," she added. "And that's why they resist us at every single level."
Congressional Democrats have characterized the bill as an effort to remove millions of Americans from voter rolls, and Senate Minority Leader Chuck Schumer, D-N.Y., called it "Jim Crow 2.0," a term used by some to describe modern so-called voter suppression laws.
During her remarks, Noem mentioned a handful of illegal immigrants who were registered to vote in various states.
"There is no room in our election system for people that aren't Americans," she said. "There is no room in our election system for fraudsters and foreign influence."
The secretary also called for Arizona to clean up its voter rolls, noting that DHS Systematic Alien Verification for Entitlements can be used to do just that. The program helps local, state and federal agencies determine the citizenship and immigration status of individuals.
"Maybe people who've passed away, people that aren't citizens, people that don't live here," she said. "That would make sure that, in your next election, when people are casting their votes, they know they're voting for the right decisions and that those votes are counted. And they're counted appropriately. And someone else didn't get to weigh in on their leadership."