Fàbregas, 27, has signed a five-year contract with the Blues after a disappointing three years at Barcelona.
"I do feel that I have unfinished business in the Premier League and now is the right time to return," Fàbregas said on the Chelsea website.
"I considered all the other offers very carefully and I firmly believe that Chelsea is the best choice. They match my footballing ambitions with their hunger and desire to win trophies. They have an amazing squad of players and an incredible manager. I am fully committed to this team and I can't wait to start playing."
Fàbregas added on his Facebook page: "Yes, everyone knows that Arsenal had the first option to sign me. They decided not to take this option and therefore it wasn't meant to be. I wish them well in the future."
Although the parsimonious Arsène Wenger was never likely to spend £30m when other areas of his squad require redressing, Fàbregas' affection for Arsenal and apparent willingness to re-join the club will disappoint their supporters.
Only in February Fàbregas told Chelsea coach José Mourinho to "shut up" following his criticism of Barcelona, and despite Mourinho's supposed suspicion of Spanish players, he has put any differences to one side and bolster Chelsea's midfield.
Fàbregas will don the No.4 shirt he wore for six years at Arsenal and is viewed as the long-term successor to Frank Lampard, who was released on a free transfer.
Manchester United pursued Fàbregas last summer, however the midfielder's reluctance to hand in a transfer request, as well as Barcelona's complex politics and a managerial change scuppered any prospects of a deal.
Fàbregas won the Club World Cup, Copa del Rey and La Liga at Camp Nou, yet his return to Barça coincided with their gradual decline and he struggled to establish himself in the first-team.
Luís Enrique, Fàbregas' fourth coach in under three years, replaced Tata Martino last month and was content with selling the midfielder.
Holders Spain begin their World Cup campaign against Holland on Friday night.