The 23-year-old underwent a medical and held talks at Celtic Park on Tuesday and the Hoops announced the deal was done on Wednesday morning.
Johansen was part of the Stromsgodset side which won the Norwegian league last year.
The player is Neil Lennon's first signing of the transfer window.
After being unveiled at Parkhead, Johansen said: "I'm very happy to be here, it's a fantastic club. I spoke to some players in the national team and everyone has good things to say about Celtic.
"I'm a passing player when the ball is on the ground. I am creative, I want to score and create goals.
"My fitness test went well but we will be patient about when I play. I haven't played since November so I don't think I will play against Motherwell on Saturday.
"This league here is better, it's a big step up. The clubs in Norway are nowhere near Celtic.
"You can see how much this club means to the supporters so I can't wait to play for them.
"Everybody wants to play in the Champions League. It's a dream when you are 13/14. I hope to help them qualify, of course.
"I will adapt. I have no worries about that. I expect it to be more physical. Celtic like to play football so I look forward to it."
On being handed the No.25 shirt previously worn by Celtic legend Lubo Moravcik, he said: "I had a few choices and the club were happy to give me this one. I'm going to try my best with the number.
"I'm the kind of player where the biggest pressure is the pressure I put on myself. I hope my football excites the Celtic fans. I am a passing player but you can't just pass, you need to create as well."