Rashbass will take up the London-based role in the summer in the newly established role of chief executive of Reuters.
He has worked for The Economist for 15 years in a number of roles, including managing director of Economist.com and publisher of The Economist. He has been group chief executive since 2008.
James Smith, chief executive of Thomson Reuters, said: "Andrew's track record as CEO of The Economist Group speaks for itself. He brings to us a rare blend of commercial acumen, sensitivity to the best traditions of quality journalism and editorial integrity. And he has a proven ability to build and lead great global business teams."
Rashbass said: "I am excited to be joining Reuters with its unmatched commitment to reporting with speed, accuracy, integrity and independence. As well as being, or in fact because they are, essential to Reuters journalism, these characteristics can also drive huge commercial success."