Police said initial indications are that the incident, in central Washington on the afternoon of October 3, was not a case of terrorism.
Police said a 1-year-old child who was in the car with the woman was unhurt.
Police briefly surrounded the black Infiniti sedan outside the Capitol complex, where the Senate and House of Representatives are located, before the driver was able to speed away.
Police said two security officers -- a Secret Service officer at the White House and a police officer at the Capitol complex -- were injured when they were hit by the car.
The motive behind the incident was not immediately clear, and police declined to release details about the woman, citing an ongoing investigation.
Other officials, however, speaking on condition of anonymity, described her as a 34-year-old mother from the northeastern state of Connecticut.
Officials said FBI agents had obtained a warrant and were searching a residence in the Connecticut city of Stamford in connection with the incident.
The Capitol and White House were put under a security lockdown during the incident, but the alerts were lifted a short time later.
The incident, which came on the third day of the partial U.S. government shutdown, sent tourists and employees running through the normally calm Capitol area.
It also further rattled the nerves of Washingtonians, coming just three weeks after a government contractor opened fire at the nearby Washington Navy Yard, killing 12 people and wounding three others before he was shot to death by police.
Washington Police Chief Cathy Lanier declined to speculate on the motive of the driver, but rejected suggestions it might have been an "accident."
"All our information that we have right now is, this does not appear in any way to be an accident," Lanier said. "This was a lengthy pursuit, there were multiple vehicles that were rammed, there was officers that were struck, and two security perimeters that were attempted to be breached. So, it does not appear in any way this was an accident."
The chief of the Capitol Police, Kim Dine, said he had no information to suggest a terrorist motive, saying it appeared to be an "isolated incident."
Officials said the Capitol police involved in the incident were not being currently paid because of the government shutdown.
But the officers have been declared "essential" staff, and were nonetheless at work.