The 23-year-old midfielder joins the Scottish champions on a three-and-a-half-year deal, having spent three years at the Marienlyst Stadion.
During his time at Stromsgodset, Johansen helped the club to their first league title in 43 years, scoring four times in 27 league appearances.
He becomes the fourth Norwegian to join Celtic, after Harald Brattbakk, Vidar Riseth and Thomas Rogne, and could make his debut against Motherwell on Saturday.
Neil Lennon's men are unbeaten in the top flight this season and sit 10 points clear at the top of the Scottish Premiership with two games in hand on nearest rivals Aberdeen.
Johansen has six international caps to his name and will be unveiled at Celtic Park on Wednesday.
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