Councillors also approved a commissioning plan to allow design, construction and project management contracts to be procured.
The MOD will be providing the remaining funding needed for the bridge and future maintenance through a Memorandum of Understanding (MOU).
The current bridge is in a poor condition and is subject to a weight restriction of 7.5 tonnes. This was introduced in December 2018 due to safety concerns after an inspection found significant defects.
The bridge, which runs over the main Bristol to Taunton railway line, dates back to 1941. It was built to provide access to the military aeroplane factory during the Second World War, using emergency wartime powers.
Since then the condition has gradually worsened with ongoing repair no longer an option. Without replacement, further weight restrictions and full closure will be likely in the next five to 10 years.
Winterstoke Road and Broadway are vital routes within Weston with many local businesses and communities relying on the routes to access their premises and homes, as well as previously being the emergency diversion route for the M5.
The new bridge, which is set to open in 2023, will also include improved pedestrian and cycling provision.