Darren Bent insisted he had to leave Aston Villa after too many "slaps in the face".
Brighton defender Greg Halford has agreed to extend his loan deal from Nottingham Forest until the New Year.
Martin becomes the Rams' second permanent acquisition, after the club confirmed the arrival of Burnley goalkeeper Lee Grant who, like Martin, is to officially join Derby from July 1.
The 24-year-old has agreed a two-year deal with the option of a further year after the Canaries decided against taking up the option of a further 12 months on his contract, which has allowed Martin to join County on a free transfer.
The 24-year-old forward initially joined the Rams on a one-month loan deal in February, making his debut as a second half substitute in the side's narrow 2-1 defeat to Watford at Vicarage Road.
Martin went on to make 13 League appearances for the Rams, scoring two goals manager Nigel Clough said: "Chris did really well whilst here on loan so we're pleased to get him signed up, and it's another important signing for us.
"He's a striker with different qualities to what we've already got, especially with his physical presence and ability to hold the ball up.
"We've admired him from afar for some time and monitored his progress closely, and once we got him in the building we knew he would fit in just fine with how we want to play and this group of lads.
"We can't wait to see him in action come the start August with a full pre-season behind him."
The Canaries Academy graduate had found it difficult to secure a regular first-team place with Norwich in the Premier League.
Since making his City debut as a substitute against Wolverhampton Wanderers at Carrow Road in January 2007 Martin went on to make 117 appearances for the Canaries, scoring 34 goals.
Twenty three of his goals came in their 2009/10 League One title-winning campaign.