Police said the accident happened Sunday night while the car was in autonomous mode. A human monitor was also behind the wheel.
Police said the woman, Elaine Herzberg, had not been using a pedestrian crossing. She was taken to a local hospital, where she died.
The US National Highway Traffic Safety Administration and the National Transportation Safety Board said they were sending teams to Tempe.
'Wake up call'
Companies including Ford, General Motors, Tesla and Waymo are investing heavily in research to develop self-driving cars, which are often characterised as the future of the industry and hailed as a way to reduce traffic accidents.
Many states across America have welcomed the tests in the hope of keeping themselves at the forefront of new technology.
However, there have been warnings that the technology is being deployed before it is ready.
Anthony Foxx, who served as US Secretary of Transportation under former President Barack Obama, called the accident a "wake up call to the entire [autonomous vehicle] industry and government to put a high priority on safety."
More than a dozen states in the US allow autonomous vehicles on the roads to some degree. Officials typically require a person to be on hand either in the car or remotely in case something goes wrong, according to the Center for Automotive Research.
The US is working on national safety guidelines for such vehicles.
Consumer Watchdog, a lobby group that has warned of the risks of autonomous cars, on Monday called for a moratorium of such vehicles on public roads, describing the accident as a "tragedy we have been fighting years to prevent".
"We hope our calls for real regulation of driverless cars will be taken seriously going forward by Silicon Valley and the Trump Administration," the group wrote on Twitter.
Uber started testing driverless cars in Pittsburgh in 2016. The ride-hailing firm has also been testing driverless cars in San Francisco, Pittsburgh, Toronto and the Phoenix area, which includes Tempe.
Carla Bailo, president and chief executive of the Center for Automotive Research, said more information about how the crash occurred is necessary before officials can say what went wrong and how the self-driving system should be improved.
She also said the fatality should be considered in the context of all accidents.