The London club confirmed on their website on Thursday that the deal had been finalised after the two clubs agreed a fee, thought to be between ?15million and ?20million, for the highly-rated 18-year-old earlier this month.
He underwent a medical at the Blues' training ground in Cobham last week and will now be available for Saturday's Premier League game against West Brom.
"Amazing. It's a dream come true," he said on the club's website.
"I dreamed to play here since I was 10 years old, thinking about shaking hands with all those players like John Terry and Frank Lampard and Didier Drogba.
"It's amazing to arrive here in a big club with big ambition. It was just what I was searching for."
Source: Team Talk
Source: Team Talk