The 18-year-old's transfer has finally gone through 10 days after he arrived in London for a medical, and he will be available to make his Premier League debut against West Bromwich Albion at Stamford Bridge on Saturday.
Lukaku was the Belgian First Division's top scorer in 2009-10 with 15 goals, and scored 16 last season to make him one of Europe's hottest properties.
The striker said: "It's a dream come true. 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's 6' 3" frame and physical style have prompted comparisons to Drogba, and he said he was looking forward to learning from the Ivorian.
"We have similar playing styles, but I am also very fast," he said. "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."
Villas-Boas has also hinted Chelsea will not be held to ransom over Luka Modric as he continues to explore alternatives to the club's number one transfer target.
Tottenham boss Harry Redknapp admitted it might be in his side's interests to sell the Croatia playmaker and spend the fee on strengthening their squad. Chelsea have this summer seen bids of £22 million and £27m rejected for the 25-year-old, who Spurs chairman Daniel Levy continues to maintain is not available at any price.
Chelsea have been linked with a £30m bid before the transfer window closes and Villas-Boas said: "I don't know if it's viable or not and to what extent we can go regarding his price because we are speaking about a high amount and nothing else basically. End of story."
He added: "Whatever comes from Harry's mind is his own opinion about the situation.
"I just have to study the rest of the market and hopefully find a solution for my team."
Villas-Boas last week revealed he was attempting to sign two midfielders before the August 31 deadline.
"That's a lot!" Villas-Boas quipped. "Why don't we dwell on the names that already have been spoken about and never came?
"Neymar, (Samir) Nasri - how many names have been linked with Chelsea in the past and they didn't come?
"In our operations, in whatever we do or not in the market, we try to keep it as silent as possible. We have been studying all kinds of options."