Transport Minister Keith Brown said the Forth Estuary Transportation Authority (FETA) would be scrapped and its 70 staff would be transferred to a new contractor.
Opposition politicians said the move meant the loss of local accountability, and they accused the SNP of centralisation and privatisation.
Liberal Democrat city councillor Phil Wheeler, vice-convener of FETA, said: “It’s the control freaks at Holyrood at work again. The bridge has been run since it was planned over 50 years ago by a joint board of local councillors.
“It’s disappointing that local accountability is being scrapped by the minister. He obviously was not paying too much attention when he met the FETA board back in June when we had an exchange of views and people of all parties stressed the value of having local members involved.”
The FETA board is made up of four councillors each from Edinburgh and Fife and one each from West Lothian and Perth and Kinross.
Mr Brown said: “I have concluded that a competitive tender exercise will be held for one contractor to manage and maintain both the Forth Replacement Crossing and Forth Road Bridge.
“This option will deliver value for money through market testing for both bridges and offers operational efficiencies through a twin-bridge strategy.”
He said the skills, knowledge and professionalism of FETA staff were greatly valued by the Scottish Government and he had assured them their jobs would be protected under transfer of employment regulations.
He promised a forum to ensure “regular and meaningful engagement” with local elected members, community councils and residents.
Lesley Hinds, Labour’s transport spokeswoman on the city council, said: “Everything is being centralised by a controlling SNP government. Why do they not trust local authorities and locally-elected members?
“The SNP government has this rhetoric against privatisation, yet that’s what they are doing – privatising the operation of the bridge and the jobs.”
She said a revamped FETA should have been given the role of operating the new crossing alongside the existing bridge.
She said: “That’s what most people would think was the sensible option.”
Edinburgh Western SNP MSP Colin Keir welcomed the guarantee to FETA staff. He said: “I have been advised that staff will remain on their current terms and conditions. Their skills, knowledge and professionalism are greatly valued, which is why their staff must continue to be part of the project.”
He argued the promised new forum would ensure a degree of local accountability.
Bridges elsewhere in the UK operate under a variety of models. At the Severn, a second crossing linking England and Wales was built by a business consortium under a public-private partnership. The Humber Bridge is run by a 22-strong board made up of representatives of the local authorities.
The Dartford crossing over the Thames is now run by Connect Plus (M25) Limited on behalf of the UK Government’s Highways Agency.