He will step down as executive chairman more than 40 years after setting up DFS in a former billiard hall in Doncaster.
Advent did not disclose the value of the transaction, but the deal is thought to be worth around 500 million. Kirkham will continue to have a substantial financial interest in the sofa manufacturer through loan notes to Advent.
He floated DFS – which now has 74 stores across Britain and employs more than 2,600 staff – in 1993 before taking it private again in 2004 for 507m.
The sale comes as the 65-year-old founder approaches retirement with no family successor in line to take over the running of the company.
He will be succeeded as chairman by Richard Baker, a former Alliance Boots chief executive and Advent partner.
Advent's British retail portfolio already includes discount chain Poundland. It also invested in clothing retailer Fat Face before selling in 2007.
Elsewhere, it has investments in Charlotte Russe, the young women's clothing business in North America, and Takko, the German fashion discount chain.
DFS has weathered the recession while rivals such as SCS Upholstery and Land of Leather have had a tougher time.
Advent director Andy Dawson said: "The company's relative size and reputation as market leader has allowed it to perform well during the downturn."
The new owner is expected to restart a store expansion programme which has lain largely dormant since the group returned to private hands.
Kirkham said: "Plans to expand DFS will continue to offer great career opportunities for everyone."
As well as its store base, the group has three manufacturing centres in Yorkshire, Nottinghamshire and Derbyshire.