Hogh had already been on Celtic's watch list in January, but he opted to stay with Bodø/Glimt to finish their Champions League campaign.
After picking up 54 goals in 103 games for Norwegians Bodo/Glimt, Hogh pens a four-year deal with the option of a further year in what is a record-breaking £11M deal.
Speaking in his first interview, he said that he realized it was the right move as soon as he was contacted this summer.
"I'm so happy," Kogh told Celtic TV. "When I came into the room and saw the shirt, I was really excited.
"The history of the club is so big. I spoke with some people and with my family about the club.
"The feeling they had, and the feeling I had in my stomach, was that this was a really good match."
Describing what Celtic supporters can expect from him next season, the striker responded: "I like to be in the box and score goals, but I also want to help my team-mates defensively and work hard for the team.
"I've trained throughout the summer and I've already played league games in Norway, so I feel ready. I just want to get started."
Hogh has spent the last two seasons playing in front of average home crowds of around 6,000 in the Arctic Circle and he now switches that for Celtic Park, a stadium 10x the size.