Lawson has held the top job at Leap since November 2010, and before that spent three years as deputy chief executive of the human rights charity Prisoners Abroad.
He is chair of the Centre for Youth Impact, which focuses on impact measurement in the youth sector.
A date for his move has not yet been fixed.
Sally O’Sullivan, chair of Turn2us, said Lawson’s reputation for inspirational leadership preceded him.
"We also value his compassionate understanding of personal dignity and his commitment to making a lasting and real difference," she said.
"Turn2us believes there are many opportunities ahead for cross sector collaboration to the mutual benefit of those in need. We know that Tom will further this ambition."
Mark Spelman, chair of Leap Confronting Conflict, said Lawson had been "an exceptional leader" at Leap.
"I will miss working with Tom but am pleased to see that he is going on to an important role helping tackle poverty in the UK," he said.
Lawson said was a privilege to have the opportunity to be able to contribute to the battle against UK poverty at Turn2us.
"I have loved working at Leap for the last eight years," he said. "It is a fantastic team of young people, trainers, staff and trustees who create such positive impact for young people through their creativity and dedication."