The Blades are reported to have spent £8million on the Republic of Ireland international, who has signed a four-year deal.
It is the second time this month United have paid out a club record fee ahead of its Premier League return.
Luke Freeman joined from Queens Park Rangers for a reported £5m, while manager Chris Wilder has also agreed a new contract.
"Callum fits into our structure perfectly as we look to Championship players who we feel can be successful at Premier League level," Wilder said of 24 year-old Robinson. "Callum is someone who was on our radar a number of years ago and he has progressed how we thought he would and hopefully there is plenty more to come.
"I feel Callum is a great age and what we have paid represents great value in the current market."
Robinson hit 12 goals in 27 Championship appearances for PNE last season.