The France international has penned a four-and-a-half-year deal with Avram Grant's Blues, who had a £12million bid rejected earlier this week.
Chelsea boss Grant was keen to seal Anelka's switch to Stamford Bridge before Saturday's clash with Tottenham, and the Israeli has got his man.
The 28-year-old striker has netted 11 goals so far this season and joined Bolton from Fenerbahce in 2006 for £7million.
A statement on Chelsea's official website read: "Chelsea Football Club and Bolton Wanderers can confirm that terms have been agreed between the clubs for the transfer of Nicolas Anelka."
The new signing is available for Saturday's clash with Tottenham after being registered with the Premier League in time.