The move was completed on Friday afternoon, Robinson having travelled to the Algarve in Portugal where the Blades are having a training camp.
Robinson, who was North End's top scorer last season with 13 goals, has signed a four-year contract with the Premier League newcomers.
While the fee is undisclosed, United have confirmed that it is a record buy for them.
It is understood that the deal is worth around the £8m mark and Robinson's former club Aston Villa will get a slice as part of a sell-on clause.
Blades manager Chris Wilder said: "Callum fits into our structure perfectly as we look to Championship players who we feel can be successful at Premier League level.
"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."
North End boss Alex Neil admitted he was 'gutted' to lose Robinson but will hope to use the money to strengthen his squad.
Neil told PNE's official website: "We are in a situation where we have been given a good offer.
"Callum has seen his future in the Premier League if that opportunity came along, which it did, and we had to try and maximise the cash intake for us, so we can try and strengthen the squad and strengthen the club.
“I am gutted Callum has left. He was a great lad about the place, but he has deserved his chance and he has worked very hard for a couple of seasons."
Robinson only left PNE's training camp at Fota Island on Thursday afternoon after United's bid had been accepted.
He travelled to Sheffield to undergo a medical and then flew to Portugal to link-up with his new team-mates.