The London club have confirmed on their website that the deal has 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.
"Amazing. It's a dream come true," he said on www.chelseafc.com.
"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."
Lukaku's record in Belgian football is impressive. He made his first-team debut at Anderlecht when he was 16 and has averaged almost a goal every two games in the Jupiler League.
His goals helped Anderlecht win the Belgian title in 2010 and he scored three times in the Champions League qualifying round last season, although the team went out before the group stage.
Lukaku, whose father Roger played for Zaire in the 1990s, has won 10 caps for Belgium, scoring two goals.
He will wear the number 18 shirt for his new team and cannot wait to get started for a side he has supported for the past decade.
"I watched the Premier League since I was nine," he said. "Every weekend I watched the games, I used to watch Jimmy Floyd Hasselbaink scoring goals and remember one against Manchester United, the volley at Old Trafford, and I decided I wanted to support this team."
"It was a good experience to play in Belgium but I wanted to take a big step to join a big club and I said from day one that I wanted to go to Chelsea. To be here now is so good, there are guys with a lot of experience at international level and they won everything, but I didn't, I don't have the same feeling as them. I just want to win something so I am very hungry to start with these guys."
Chelsea failed to lift any silverware last season but Lukaku is confident that will be rectified this term.
"I understand there is a lot of pressure on the team but with the players that we have, so much experience and so much talent, and with a great boss also, you can see that we can be confident we can make a successful season," he said.
On his targets for the campaign ahead, he added: "First of all I just want to learn and train as much as possible to improve myself and show the coach that when he needs me I am ready.
"Then I hope to make minutes for the team and be helpful for the team, because the team is the most important thing. If you think positively about the team it will be positive for you."