Bristol City Council, Bath and North East Somerset Council and South Gloucestershire Council voted for devolution.
Four councils were offered a Â£900m deal over 30 years to invest in key areas, in the Chancellor's March Budget.
Earlier this month, North Somerset voted against the plan, saying the deal was "not attractive enough".
The remaining three councils held cabinet and full council meetings to decide on whether to accept the deal or not.
Before the meetings, some councillors had told the BBC that due to the uncertainty over Brexit, the deal and funding pledge may not happen.
The funding may not be realised if the country slides into recession or, if a general election is called a new government may not honour the offer, councillors have said.
The deal, made in the Chancellor's March budget, includes creating a regional mayor, sometimes referred to as a Metro Mayor.
The councils would be given more power over key areas such as transport and roads planning, as well as housing and would receive Â£900m over 30 years.
The three councils will now need to renegotiate the deal as it was based on a four-council agreement.
North Somerset Council declined the offer earlier this month, fearing it would add a costly layer of bureaucracy into decision-making.
Some parish councils and residents under the local authority also said they were worried it could be a return to the former Avon authority with places like North Somerset overlooked for the needs of the larger cities.
Three councils have voted to accept a single directly-elected "Metro Mayor" for the west of England.