Linney, the joint chief executive of Outsourcery, previously featured on TV in the Channel 4 show Secret Millionaire.
Linney will join regulars Duncan Bannatyne, Peter Jones, Deborah Meaden and another new Dragon, Kelly Hoppen.
In the BBC Two show, business leaders hear pitches from entrepreneurs before deciding whether to invest.
Linney said he was "excited" and would "widen the appeal of the Den to information economy entrepreneurs," adding, "they are key to the future success of the UK economy."
The qualified solicitor also worked as an investment banker and venture capital fund manager and was named in The Power List of the UK's 100 most influential black Britons.
His company Outsourcery was launched in 2007 and employs 150 people across the UK.
Executive producer Ceri Aston called Linney's story "inspiring", adding: "Piers is a new breed of dragon. A young and dynamic digital entrepreneur, he has his finger on the pulse of an industry that has never been represented in the Den before.
Departing dragon Paphitis said "The time felt right to give up my seat, stop breathing fire and allow someone else to enjoy the wonderful experience of being a Dragon."
Haulage firm boss Hilary Devey left the show last year for her own Channel 4 series and interior designer Kelly Hoppen was announced as a new dragon last month.
Cloud computing pioneer Piers Linney has been named as the new member of the Dragons' Den, replacing Theo Paphitis.