The firm picked up the job with a cut-price bid of £49.8m on the building, which was initially estimated to cost £80m by the client.
Balfour Beatty beat BAM Construct, Bowmer & Kirkland, Interserve, Miller Construction and Sir Robert McAlpine.
Sources at the losing bidders said they were surprised by the winning price.
A University of Sheffield spokeswoman said the overall cost of the project was still £80m with the £49.8m representing just the value of the construction contract.
The university gained planning approval to build the engineering faculty late last year despite some local opposition because the project will see the Edwardian wing of the former Jessop Hospital for Women demolished.
But problems with the site look set to rumble on as national heritage charity SAVE has vowed to launch legal action against the council to halt the clearance of the site.
It could take months for any case to be heard by a judge, delaying demolition.
The university said it examined potential alternatives to demolition, and considered several other locations, but the preferred choice was the best for the university, the city and the region.
Architect RMJM designed the building and is joined on the project team by cost consultant Turner & Townsend and Arup as M&E consultant.