Vieira, 47, succeeds Frederic Antonetti, who was dismissed early last week after Strasbourg finished 15th in Ligue 1, two places and five points above the relegation zone.
"I welcome Patrick Vieira, whom I know from the French national team; I am very happy to welcome him to Strasbourg," said Marc Keller, the Strasbourg president and a former France midfielder in a statement on the club website.
"He corresponds to the profile we were looking for: a manager with international experience, and who also has a good knowledge of Ligue 1 and its young players." The Alsace club is under new ownership, with The BlueCo consortium, led by Chelsea owner Todd Boehly and US-based Clearlake Capital, buying a majority stake on June 22.
He has been head coach at New York City FC, Nice and Crystal Palace. He said he was "particularly happy" to take over at Strasbourg.