O'Neil has been out of work since his sacking at Wolves in December 2024 and the 42-year-old Englishman will be in charge for Saturday's French Cup match away at Avranches.
"I am proud to join this incredible club and I can't wait to get started," O'Neil said in a statement.
"Racing has a unique history, extraordinary passion, great resilience and, of course, loyal supporters who want to see this team play beautiful football and succeed.
"We have a group of high-quality players and clear, ambitious goals for the season. My priority is to work hard with the team and give my all for the club's success."
Rosenior was announced as Chelsea's new coach on Tuesday, making the switch from Strasbourg to the Premier League side who are both part of the same multi-club ownership structure.
The French club was taken over by the BlueCo consortium in 2023 and took Rosenior to London to replace Enzo Maresca.
Maresca parted ways with Chelsea on New Year's Day after a poor run of form, winning one of their last seven league games to lose ground in the title race, and a falling out with the club owners.
O'Neil joined Bournemouth's backroom staff in 2021 and was appointed caretaker manager in August 2022 after the sacking of Scott Parker. He was made permanent coach in November but was sacked at the end of the season despite avoiding relegation.
In August 2023, O'Neil signed a three-year contract at Wolves and led them to a 14th place finish in the Premier League in his first season. But he was sacked in December 2024 after two wins from 16 games left them in the relegation zone.
Strasbourg are seventh in the Ligue 1 standings on 24 points after a run of five games without a win.