It is a record fee for United, who despite only being promoted to the Premier League this season, remain in contention for the European places.
On pulling off what many see as a coup for the Yorkshire club, manager Chris Wilder said: "The board have really backed us in this window to bring in players of the quality of Sander," Chris Wilder told United's official website.
"He fits the bill in so many ways. He's a Norwegian international who has played 20 times for his country. He's also played Champions League football and we're delighted he has decided to come to Bramall Lane.
"Sander is someone we have known about for quite a while now. We have tracked him over a sustained period of time, he was a player we highlighted as someone who could come in and really improve us and we are thrilled he's agreed to come here.
"When we sat down with Sander it was clear that he was really excited about coming here and that was great to see.
“I think this signing just shows how far we have come in a short space of time, that we are now attracting the likes of Sander to the football club. It's a brilliant signing for us."
Velkommen, Sander ✍️ pic.twitter.com/rz1HZl09Pm
— Sheffield United (@SheffieldUnited) January 30, 2020