The Stirling-based firm has secured the job with a bid of £18m ahead of rivals Balfour Beatty, Barr, Graham and Kier.
University buildings will be overhauled at the Cowcaddens Road site to create a 500-seat conference facility, a new glass reception and atrium and teaching rooms.
Two buildings, the George Moore and Hamish Wood, will be significantly renovated and will be better connected to the campus centrepiece, the award-winning Saltire Centre.
Construction work is due to start in June and is expected to be completed by January 2016. Glasgow-based architects, Page\Park are behind the design of the project.