Mr Budd, along with leaders from Stockton, Redcar & Cleveland, Hartlepool and Darlington, signed the deal with Chancellor George Osborne in October, subject to the agreement of the five councils.
A full meeting of Middlesbrough Council has now backed the deal with Central Government for a range of devolved powers by a majority of 34-4, with one abstention.
Addressing the meeting, Mr Budd said: “If somebody says ‘do you want more power over what happens in your area, more influence over what happens in your area and more money for your area’, there is only one answer.
“I think it is worth saying that we in the Tees Valley have been working together in one form or another for two decades now, more so than anywhere in the country except perhaps Manchester.
“The deal fits with economic aspirations we have already for what is effectively an economic area – the Tees Valley.
“It gives us powers over a whole range of things from investment through European funding, planning and housing, transport projects, education, skills and culture and others as well.
“This isn’t the final version, there are a lot more stages to go through but this is the outline. Despite any natural reservations this is something we all felt convinced was an opportunity for our part of the world.”