Former France international Vieira coached Premier League side Crystal Palace from 2021 to this year. He was sacked mid-March following a 1-0 defeat at Brighton & Hove Albion that left the south London club three points above the relegation zone.
"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 the 2022-23 league season.
Vieira won the 1998 World Cup and 2000 European Championship as a player with France. He earned more than 100 caps for his country, scoring six goals.
At club level, he helped Arsenal to three Premier League titles, and won Serie A four times - with AC Milan in 1995-96 and with Inter Milan in three consecutive seasons from 2006-07.
The Frenchman began his managerial career with his former side Manchester City's under-23 team from 2013-2015, before he went on to take over at New York City FC followed by a 2-1/2 year stint at Ligue 1 club Nice.
"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."
