Saha replaces Patrick Griffiths, who left in March to set up the digital shop Work Club alongside the agency's executive creative director Andy Sandoz and Martin Brooks, the chief executive of the Omnicom-owned Zulu network.
Saha, who will join the agency at the beginning of November, will be responsible for the planning strategy across the agency's clients, including O2, Mercedes-Benz, Unilever and Sony PlayStation. He will also join the agency's board.
Before joining Xbox in 2004, Saha spent four years at Adidas, where he was the global ad and media manager. His career includes stints at JWT. In his new role, he reports to Jane McNeill, the Agency Republic managing director.
McNeill said: "Indy is the perfect hire for Republic. His client-side experience, coupled with his passion for creativity and emerging technologies, make him incredibly well placed to take Republic to the next stage."
Saha is the latest in a line of traditional agency and client people to move into digital. Earlier this year, David Pattison left PHD to join ILG, i-level's parent company; Alan Rutherford quit Unilever for Digitas Global; and Giles Montgomery, a creative at Wieden & Kennedy Amsterdam, became the creative director at Grand Union.