The 18-year-old has scored six goals this season, including one in the Scottish champions' 2-1 win over Barcelona in the Champions League.
The striker told the club's website: "I'm delighted to have signed this new contract. I'm a Celtic fan and this is where I want to play my football.
"The last couple of weeks have been incredible, and to sign the contract just a week after scoring against Barcelona is brilliant.
"I know I've still got a lot of hard work ahead of me in my career, but it's great to know that I'll be here as part of this exciting young team the manager's building at Celtic."
Watt joined Celtic on a three-year deal from Airdrie United in January 2011.