Vieira was recently in charge of Crystal Palace, where he spent 20 months, before getting sacked in March due to poor form.
The former France international led Palace to 12th position and the FA Cup semi-finals in his first season in charge.
Vieira has coached in Ligue 1 before at Nice between 2018 and 2020, and has also managed New York City FC in America’s Major League Soccer.
"Patrick's arrival should allow us to make further progress and to take a step forward," said Strasbourg President Marc Keller.
"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."
Strasbourg finished 15th in Ligue 1 last season.
"I am particularly happy to join Racing," Vieira told the club website.
"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.
"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."
Vieira, who won the 1998 FIFA World Cup and European Championship in 2000 with France, was captain of the ‘Invincibles’ Arsenal side who went the entire 2003/2004 Premier League season unbeaten.
