The 20-year-old defender was born in Holland and moved to England as a child, beginning his professional football career with AFC Wimbledon.
He joined Wednesday in 2019, making 21 appearances and catching the eye of the Scottish Premiership outfit, who have also signed Liam Shaw from the Yorkshire club this summer.
Postecoglou told Celtic’s official website: “It’s great. He’s someone that the club’s been tracking for a while and, when I came on board, I managed to have a meeting with him and he’s a good young man, he’s very ambitious.
“He’s just beginning his football career, but he shows all the attributes on and off the field to want to become a top player, so it’s great to get him here.
“He can play a couple of positions at the back and, most importantly, he’s really keen to learn and develop and I think this will be a great environment for him.
“He’s a modern defender. He’s good athletically, he’s mobile, he’s quick and he wants to get on the ball.
“He can play a couple of different positions and, from our perspective, and the way we want to play our football, there are certain things I look for in players irrespective of age, and he’s got some really good attributes to be a top defender.
“The most important thing is, he’s got a real desire to reach the very top. He’s excited about joining the club, which is important to me because that’s the first thing I want to hear from players, that they’re really keen to join us.”
On his official Twitter account, Urhoghide said: “Thank you Sheffield Wednesday and all those involved over the last two years from staff to fans.
“Since my debut at Brighton away to the Yorkshire derby win at Leeds, it’s given me memories that will last a lifetime.”
The Owls have signed 19-year-old centre-half David Agbotnohoma from Southampton and Leojo Davidson, 17, the former Manchester City right-back.