Construction on the new building is expected to begin next April and create an estimated 500 construction jobs.
Building plans attracted some local opposition because the project will see the Edwardian wing of the former Jessop Hospital for Women demolished.
But Sheffield Council’s planning meeting was told the new building will help the university cope with an expected increase of 1,600 engineering students by 2020 and will put £20.6m into the Sheffield economy every year.
Architect RMJM designed the striking building and is joined on the project team by cost consultant Turner & Townsend, consulting engineer Arup and M&E consultant Buro Happold.
University vice-chancellor Professor Keith Burnett said: “We recognise that we have been given a wonderful opportunity to create a fitting legacy for the future, an opportunity we take very seriously indeed.
“We are committed to investing in a high quality, stunning building which will not only be the centrepiece for the Faculty of Engineering’s continued success, but will also be a source of enormous pride for the city.”