The former champions had earlier gathered outside the Circuit de Catalunya garage for a team photograph to mark Venezuelan Pastor Maldonado's first career victory and their first in nearly eight years.
Team founder and principal Frank Williams, who is in a wheelchair, had been in the garage when it filled with smoke and flames but was quickly wheeled to safety, his daughter Claire said.
The blaze was fought by staff from Williams, Caterham and Force India from the pit lane side before circuit fire services arrived.
The Grove squad confirmed four members of the team were injured in the incident. One has already been released while the other three are now receiving treatment at local hospitals.
"The Williams F1 Team would like to thank all of the teams and the FIA for their support in today's incident," said the team in a statement.
"The team, the fire services and the police are working together to determine the root cause of the fire and an update statement will be released in due course," it added.
It is so far confirmed only nine members were from F1 teams but other team personnel were understood to be involved.
Caterham said four of its members had been taken to the medical centre for examination, while another Force India mechanic had also suffered from smoke inhalation and was under observation.
Formula One's ruling body, the FIA, said a total of 31 people were seen by medical staff from the circuit. Seven had been transferred to local hospitals.
"The FIA is collaborating closely with the Spanish authorities investigating this incident and will be providing a further update as soon as more information becomes available," said the FIA.
Caterham, who have an adjacent garage to Williams, said four of their staff had been taken to the medical centre for examination - one with a minor hand injury and three with respiratory issues.
Force India deputy team principal Bob Fernley said one of his staff had suffered smoke inhalation.
The fire, with thick, acrid black smoke billowing out of the garage and across the paddock, started about 90 minutes after the end of the race.
Maldonado was photographed carrying his 12-year-old cousin piggyback through the smoke, with the youngster clutching the winner's trophy.
Photographs also showed Brazilian Bruno Senna's car in the middle of a gutted and charred garage. Maldonado's had been parked elsewhere under post-race parc ferme rules.
Team officials and guests were ushered to safety and the area cordoned off. A medical helicopter was seen to take off after a while.
The fire was the second in the Formula One paddock in five races this season, with one breaking out in the Lotus hospitality area at the Malaysian circuit in March.