Keppie Design, based in West Regent Street, had already been working on the blueprint.
Building work at the Monklands Hospital site, Airdrie, is due to begin later this month.
The architects have designed a centre with a "double height" forecourt-style entrance with treatment departments across the ground floor.
There will be lots of natural light from a courtyard at the centre of the building .
Keppie chiefs are working with construction company Laing O'Rourke on behalf of NHS Scotland.
Peter Moran, Keppie's public sector director, said: "It will be a fantastic facility and we are honoured to be part of the project team right from inception through to completion."
The new Beatson will handle up to 80 patients a day. Initally, these will be from Lanarkshire, but over time the new building could also treat people from Glasgow's East End and the Forth Valley health area.