Working with Dutch steel specialist Hollandia and designer Ramboll, Graham beat two rival bids from Balfour Beatty, and Roadbridge in JV with American Bridge.
The 184m-span swing bridge between Renfrew and Yoker near Clydebank forms the centre-piece of a major infrastructure project to transform the Clyde waterfront.
All planning consent is in place and construction is scheduled to start in Spring and take three years to complete.
The contract will be finalised later this month following conclusion of the procurement process.
The Clyde Waterfront and Renfrew Riverside project is being jointly funded by the UK and Scottish governments through the £1.13bn Glasgow City Region City Deal.
A new road will be constructed through Renfrew connecting the bridge to the Advanced Manufacturing Innovation District Scotland (AMIDS) being developed by the local authority, in collaboration with the Scottish government and Scottish Enterprise.