Nesta said in a statement that Ravi Gurumurthy, who has spent six years at the New York-based IRC, will succeed Mulgan in December.
Gurumurthy was previously a speechwriter for David Miliband, who is now chief executive of the IRC, while Miliband was Foreign Secretary and at two other government departments.
Gurumurthy, who is the brother of the Channel 4 news presenter Krishnan, also spent time as director of strategy and communications at the Department of Energy and Climate Change.
Mulgan, who has been chief executive of Nesta for more than eight years, has not yet revealed his next role after his departure.
He had intended to leave in 2016, a blog published on the Nesta website in January said, but Mulgan said he postponed his departure for undisclosed reasons.
Nesta did not reveal what Gurumurthy’s salary would be, but said it would be released in forthcoming annual reports.
Mulgan was paid £156,431 in the year to 31 March 2018, Nesta’s latest accounts show.
Sir John Gieve, chair of Nesta, said: "I am delighted to welcome Ravi to Nesta. His record of strategic thinking and innovation will help him build on Geoff Mulgan’s achievement in transforming Nesta from a quango to an independent charity promoting innovation for social good.
"Under Ravi, I am confident Nesta will continue to nurture the best innovations, show that they work and collaborate to scale."
Gurumurthy said: "Nesta plays a unique role in generating, researching and scaling innovation in the UK and globally.
"I’m thrilled to have the opportunity to build on the terrific work done by Geoff Mulgan and the team, and harness the skills, networks and resources of Nesta to tackle the biggest societal challenges of our time."