Brewer, citing what she described as a federal failure to secure Arizona's southern border, signed a broad immigration crackdown into law in 2010 to try to crack down on the flow of illegal immigrants into the state where an estimated 360,000 undocumented people live.
Critics of the law, which includes a requirement that police check the immigration status of anyone they stop and suspect of being in the country illegally, have said it could lead to racial profiling.
The shooting took place near the border town of Naco, southeast of Tucson, which remains a corridor for smuggling marijuana and people, despite the construction of a tall, steel fence along the border.
"We need to redouble our efforts to secure the border and ensure the safety of Border Patrol agents," U.S. Democratic Representative Ron Barber, who represents the southern Arizona district where the shooting occurred, said in a statement.
Sheriff's deputies were called to the scene at 1:33 a.m. local time (4:33 a.m. EDT/0833 GMT) and found one agent dead and another with non-life-threatening injuries, Capas said. A third was unharmed. FBI agents were also investigating.
Across the border from Naco in a Mexican town of the same name, Mexican police said a team of soldiers and federal and local police was searching for a suspect or suspects in the case. "We have no information about anybody being detained," said a Naco police officer who declined to be named.