The player, who spent a brief spell at the club on trial last pre-season - scoring in a friendly against Rangers- has signed a three-year deal.
He agreed to join the Iron to Laws and chairman Steve Wharton in a meeting on Monday evening.
Sharp scored nine goals in 16 games at Rushden last season, virtually keeping them in the Football League, and has scored seven times in pre-season for the Blades.
He has been prolific at youth level for Neil Warnock's side and will increase the competition up front, giving his new boss Laws plenty to ponder as they prepare for Saturday's visit to Nottingham Forest.