The measure takes effect 90 days after the current legislative session ends, which likely puts the effective date in July or August.
"I believe this legislation not only protects the Second Amendment rights of Arizona citizens, but restores those rights as well," Brewer, a Republican, said in a statement.
Alaska and Vermont now do not require permits to carry concealed weapons.
By eliminating the permit requirement, the Arizona legislation will allow people 21 or older to forego background checks and classes that are now required.
Supporters say the bill promotes constitutional rights and allows people to protect themselves from criminals, while critics worry it will lead to more shootings as people with less training have fewer restrictions on carrying weapons.
Some police officials are concerned the law will lead to more accidental gun discharges from people untrained in firearm safety, or that shooters in stressful situations will accidentally strike innocent bystanders with stray bullets.
"I know a lot of 21-year-olds; the maturity level is gravely concerning sometimes," said El Mirage Police Chief Mike Frazier, an Arizona Association of Chiefs of Police board member. "If you're going to be carrying a weapon you should know what the law is and how to use it."
However, the measure was supported by police unions representing rank-and-file officers, who said their best friend on the streets is a law-abiding citizen equipped to protect themselves or others.
The police chiefs group initially opposed the bill but then took a neutral stance after some provisions were changed at their request. Brewer's office also participated in negotiations on changes to the bill.
A Democratic leader, Rep. Kyrsten Sinema, of Phoenix, said the bill deprives law enforcement of a tool "to separate good guys from the bad guys." With a permit requirement, police encountering a person with a concealed gun but no permit had reason to suspect that person was not a law-abiding citizen, she said.
The Arizona Citizens Defense League, a gun-rights group that lobbied for passage of the "constitutional carry" bill, said gun owners foregoing permits still should get training. "The heaviest thing about wearing a firearm is the responsibility that comes with it," the group said.
More on Gun Rights
Arizona's permissive gun laws gained national attention last year when a man openly carried a semiautomatic rifle to a Phoenix protest outside a speech by President Obama.
Nearly all adults can already carry a weapon openly in Arizona, and supporters of looser laws argue that gun owners shouldn't face additional restrictions just because they want to hide the weapon.
Currently, carrying a hidden firearm without a permit is a misdemeanor punishable by up to six months in jail and a fine of up to $2,500.
Forty-five other states require permits for hidden guns, and two states - Illinois and Wisconsin - prohibit them altogether.
Federal law requires anyone buying a gun from a licensed dealer to undergo a background check, but that requirement does not apply to sales by individuals who aren't dealers. Arizona's law won't change that.
Under the Arizona legislation, people carrying a concealed weapon will be required to tell a police officer that if asked, and the officer can temporarily take the weapon while communicating with the person.
More than 154,000 people have permits to carry a concealed weapon in Arizona.
The bill acted on by Brewer was the first attempt to lift the permit requirement to reach an Arizona governor's desk.
Brewer's predecessor, Democrat Janet Napolitano, in 2007 vetoed two related bills. One would have reduced penalties for carrying a concealed weapon without a permit. The other would have allowed a person without a permit to carry a gun largely concealed as long as any part of it or its holster was visible.
Brewer in 2008 signed into law a bill allowing a person with a permit to take a gun into a restaurant or bar serving alcohol as long as the establishment doesn't prohibit it and the person isn't drinking alcohol. Napolitano vetoed a similar bill in 2005.