The 18-year-old has been rewarded for his impressive form, which included the decisive goal in last week's 2-1 Champions League win over Barcelona.
The club revealed the new deal would tie him to the club until summer 2016.
Watt made an instant impact in the first team in April when he struck two goals within six minutes of coming on for his debut in a 3-0 victory over Motherwell at Fir Park.
The teenager, who has six goals this season, said on the club's official 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 has scored a total of eight goals in 17 appearances for Celtic, 10 of which came off the bench, since signing from Airdrie last year.