The London club confirmed on their website 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 Barclays Premier League game against West Brom and hje told Chelsea's official website: "Amazing. It's a dream come true."
He added: "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."
Lukaku caught the eye of a number of Europe's top clubs after his free-scoring efforts for Anderlecht, but made it clear he was determined to move to Stamford Bridge.
His power, pace and eye for goal have seen him likened to new team-mate Didier Drogba.
While flattered by such comparisons, Lukaku believes he also offers something different and can learn a lot from the 33-year-old Ivory Coast forward.
"It's nice, I like it a lot," he said on www.chelseafc.com.
"We have similar playing styles, but I am also very fast. I want to learn off him how it is to play here at the Bridge and how to prepare for games. I can also learn off the other strikers and experienced players in the squad. I think if you are clever on and off the pitch you will get accepted and that's what I want."
Source: PA
Source: PA