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The 41-year-old Strasbourg boss told a press conference earlier in the day at the Ligue 1 club that he had "verbally agreed" on the move.
Chelsea officially announced that Rosenior, who will take charge of his first match in the FA Cup at the weekend, had signed a deal until 2032.
"I am extremely humbled and honoured to be appointed head coach of Chelsea Football Club," he said in a statement on the Premier League team's website.
"This is a club with a unique spirit and a proud history of winning trophies.
"My job is to protect that identity and create a team that reflects these values in every game we play as we continue winning trophies."
Rosenior has just three years' experience as a manager, with Hull City and French club Strasbourg, and has never coached in the Premier League.
But he had been widely touted as the frontrunner to succeed Maresca since the Italian left Chelsea last week.
Strasbourg are part of the Todd Boehly and Clearlake Capital-owned BlueCo multi-club ownership group that also controls Chelsea.
Rosenior is Chelsea's fourth permanent managerial appointment since BlueCo took control of the club previously owned by Roman Abramovich in 2022.
The club said the Englishman had shown "he can build teams with a clear way of playing while setting the highest standards with players on and off the pitch".
"While there will continue to be a focus on player development, the club's expectations and ambitions remain high," Chelsea added.
Rosenior said he had wanted to announce the news himself in Strasbourg because of his deep feelings for the French club.
The former defender, whose father, Leroy Rosenior, was also a player and a manager, told reporters the chance to manage Chelsea was "an opportunity I cannot turn down".