Except as otherwise provided in division (D) of section 713.23, and sections 125.04 and 5513.01 of the Revised Code, no contract in the department of public service or the department of public safety, in excess of five thousand dollars, shall be awarded except on the approval of the board of control of the city, which board shall direct the director of the appropriate department to enter into the contract. The members of the board shall prepare estimates of the revenue and expenditure of their respective departments to be submitted to the city legislative authority by the mayor, as provided by law.