The Republic of Ireland international signed a new contract with Norwich in February 2023 to extend his stay until 2028.
However, Idah has struggled to hold down a place in David Wagner's side this season, having started just 14 of his 28 Championship appearances.
Despite having to settle for a substitute role on a number of occasions, Idah has still gone on to register six league goals this term.
Upon his arrival, Idah told Celtic TV: "It's such a huge club with such a big history behind it and I can't even put into words what it's like to be at Celtic.
"My family and my friends are all delighted that I'm here and I just can't wait to get started. All of us back home support Celtic so it's huge for me and I'm so excited to be here.
"I've got a lot of experience in first-team football. I've been in the first-team now since I've been 17 or 18, so it's not new to me.
"But coming here, it's exciting with the fans here it's going to be really good and I just can't wait to play in front of them."
The striker, who has made 20 international appearances, revealed he spoke to his Irish teammates Liam Scales and Mikey Johnston before completing the loan move to Celtic Park.
Idah added: "I've spoken to both of them and they spoke massively about the club and how great it is, the fans and the city, so they helped me make my mind up as well."
The forward will be unable to play alongside Johnston at Celtic after the winger completed a loan move to West Bromwich Albion earlier on deadline day.
The 22-year-old becomes Celtic's second signing of the winter window following the arrival of Nicolas Kuhn from Rapid Vienna.
Celtic, who currently sit top of the Scottish Premiership table, will be in action Saturday when they travel to Pittodrie for a clash against managerless Aberdeen.