The Blues winger has been biding his time over the offer, with the club eager to fend off interest from Bundesliga giants Bayern Munich.
The Bavarians came close to signing him in January, but Chelsea rejected several bids for the 18-year-old.
Hudson-Odoi suffered an Achilles injury in April, ending hopes of a summer departure, and he has now put pen to paper on a five-year deal.
Reports suggest the contract is worth £120,000 a week and Chelsea are eager to see him develop into a future England international.
Despite handing in a transfer request in January, Hudson-Odoi is now happy to stay put in west London after Lampard's arrival in the summer.
Chelsea's record goalscorer has assured him over his game time and the youngster says he has already learned a lot from the club legend.
He told Chelsea's official website: "I've been thinking for a while what it would be like to be playing for Chelsea for the next five years and how everything might go, especially now Frank is here as the manager.
"It's a great feeling for all of us because we know he gives everybody a chance if you're playing well and working hard. He's a very good manager.
"I can tell that by what he's trying to do and the things he's trying to implement into our football. He has the mentality of winning, which we all want and we all need, so him coming in was just an extra boost for me. He's already given me confidence, having a chat with me here and there, and it's great when a manager has faith in you."
Hudson-Odoi follows fellow midfielders Ruben Loftus-Cheek and Mason Mount in signing new contracts and he could return to action in Wednesday's Carabao Cup clash with Grimsby Town.