Hooper, 25, has been a long-term target for the Canaries, with manager Chris Hughton seeing several bids rejected in the January transfer window and he had looked set for a move to Championship side QPR, with the Hoops understood to have agreed a fee in the region of £5.5m.
But Norwich stepped in with a similar offer and the player has agreed personal terms on a contract that will keep him in Norfolk until 2016, with an option for a further year.
Hooper, who began his career at Tottenham before spells at Grays, Southend, Leyton Orient, Hereford and Scunthorpe, joined Celtic in July 2010 in a £2.4m deal, scoring 82 goals in 137 appearances, winning the SPL Golden Boot in 2011-12.
He becomes the seventh arrival at Carrow Road this summer and speaking to Norwich's official website, he said: "There's been a lot of speculation going back to January and I'm pleased to have finally put pen to paper and joined Norwich.
"I can't wait to get down to work, to get training with the lads and working with the coaching staff. I knew Chris Hughton and Colin Calderwood from my youth days at Spurs so it will be great to be back working with them in the Premier League.
"I have scored goals at every level I have played at and the challenge of playing in the Premier League is really exciting to me."