Patrick Vieira, the Arsenal midfielder, 47, had been out of work since being sacked as boss of Premier League side Crystal Palace in March.
“I am particularly happy to join Racing,” said Patrick Vieira.
“I know the history and the identity of this club, the fervour it arouses, the importance it has for its region, which is a place of football and passion.”
Chelsea’s owners recently agreed a deal to buy a majority stake in Strasbourg, who finished 15th in the French top flight last season, five points above the relegation zone.
It is not the first time that Vieira has worked in a multi-club group. He coached Manchester City under-23s before taking over at New York City FC, who are also part of the City Football Group.
He will replace Frederic Antonetti, 61, who guided Strasbourg away from relegation last term but agreed to part ways following the recent investment.
Vieira said: “The rebuilding of the club in recent years, under the direction of Marc Keller (Strasbourg president), is an example to be admired.
“Working with him is a real pleasure for me. Today, a new cycle begins for Racing and it is exciting as a coach to be able to build something while relying on the values that have defined the strength of the club.”
Patrick Vieira began his coaching career when he retired from playing with Manchester City in 2011, and took charge of the club’s elite development squad in 2013.
In 2015, the Frenchman made the switch to New York and the MLS before returning to Europe three years later with Ligue 1 side Nice.
Vieira led Nice to a seventh-place finish in his first season but was sacked in December 2020 after a run of poor form.
In July 2021, a decade after he retired from playing in the English top flight, Vieira returned to the Premier League as Palace manager in place of Roy Hodgson.
He guided the Eagles to 12th in his debut season, while Palace also reached the semi-finals of the FA Cup.
However, Vieira was dismissed in March, with his team winless in 2023 and only three points above the Premier League relegation zone.