Grabban has agreed a three-year deal, with an option for a further year, and arrives at Carrow Road after scoring 35 goals in 93 appearances in a two-year spell with Bournemouth.
Grabban was the Cherries' leading goalscorer last season with 22 goals as Eddie Howe's side secured a 10th-place finish in the Championship.
Grabban told Norwich's official website: "It's good to start a new chapter, get going again with Norwich and hopefully get promotion.
"I've played here before and the atmosphere's been great, the fans are great. It's exciting to be a part of the club now and hopefully do well for them."
Norwich boss Neil Adams said: "I am delighted that Lewis has put pen to paper to become my first signing.
"He will provide pace, intelligent movement and a high work ethic to the front line, and of course he showed in the Championship last season that he is capable of scoring a variety of different goals.
"Lewis possesses precisely the type of attitude, application and quality that I expect from my players, and I am really looking forward to integrating him into the squad."
Grabban started his career as a trainee at Crystal Palace, before loan spells at Oldham and Motherwell saw him secure a move to Millwall and later Brentford.
He moved to Rotherham in July 2011, where he went on to make 47 appearances in all competitions and scored 21 goals, before heading to Dean Court in the summer of 2012.