Robb has led the housing and homelessness charity Shelter for seven years. Before that he was the first director general of the Office of the Third Sector, the precursor to the current Office for Civil Society.
Before joining the OTS he was director of public policy at the National Council for Voluntary Organisations.
The JRF and the JRHT said Robb would be paid approximately £150,000 a year to lead the charities. The JRF had an income of £5.7m and an expenditure of £14.7m in the year to 31 December 2015.
Robb said Unwin had been an inspirational leader and would be a hard act to follow.
"Under her leadership the organisation has developed a comprehensive set of proposals that should inspire all of us to work together to finally bring an end to poverty in the UK," he said.
"Our challenge will be to build the partnerships across the political parties, across nations and across sectors that can make this a reality over the coming years."
Tony Stoller, chair of the JRF and the JRHT, said: "The organisations are looking forward to Campbell’s leadership at a time of extraordinary change and opportunity.
"He brings a strong track record and commitment to our vision for a prosperous and poverty-free UK.
"Julia has been an outstanding leader of the JRF and the JRHT during her 10 years as chief executive and has overseen major changes in our research, policy, housing and care work.
"She leaves both organisations in a strong position to build on their achievements under her leadership and take them to the next stage of their development."
Derek Myers, chair of Shelter, said Robb had been at the heart of Shelter's work for the past seven years and would be a "brilliant leader" at Joseph Rowntree.
The JRF focuses on creating social change through research, policy and practice, while the JRHT is a registered housing association and care provider in York and north-east England.