The 18-year-old has committed himself to the Parkhead club until summer 2016 after his decisive goal in the 2-1 Champions League win over Barcelona last week helped move negotiations along.
The composed finish was far from the only evidence that Watt could have a significant long-term impact at Celtic Park but it has heightened interest in the forward, who signed from Airdrie last year in a deal worth about £80,000.
However, the teenager does not appear to be in danger of losing his grounded approach to his life and career.
"Everybody is stopping and talking to me and saying 'well done' for the Barcelona match," Watt said in a media conference at Celtic Park after signing his new deal today.
"Ever since then I've been happy. I've just taken it in my stride.
"It has happened so I can't dwell on it. It will be a good memory for the rest of my life but I just need to get on with the rest of my season and maybe look back at the end of the season.
"You don't want to go about bragging about it. It's obviously a big thing but other people have done it before and you need to take it in your stride.
"You don't want to be fazed by it and let it get in your head and start thinking you are better than what you are.
"It's a goal but I have got a lot more to do in my career."
Watt dismissed suggestions the goal would put pressure on him to keep achieving.
"All the fans are happy I scored," he added. "They are not putting pressure on me, they are just delighted that it's just a big deal for the club.
"Hopefully I can emulate that and help us go through to the last 16."
Watt's promise was evident from the moment he made his Celtic debut.
The teenager, whose pace, strength and skill offer him the potential to be an all-round striker, scored two goals within six minutes of coming on for his Celtic debut against Motherwell at Fir Park in April.
He has also scored six times this season, including one in each of Celtic's past three games.
But he said: "I will never get ahead of myself and I just need to keep my feet on the ground, and I will do.
"I have got to deal with the pressure of people saying I am going to be the next thing.
"I have just got to keep going and doing what I do and keep the hunger. I will always have the hunger when I play for this club.
"You never really believe you have made it. I have still got a long, long way to go to keep my place in the team and hopefully I can do that over the next two or three years.
"In football you are always learning. You are never going to be the finished article unless you are the likes of Messi or maybe Ronaldo."
Watt came into organised football later than most but soon made up for it by scoring goals for Airdrie's first-team after his potential was spotted by Diamonds boss Jimmy Boyle.
"A few people around me like Jimmy Boyle told me I had the ability to go on and do stuff like this," Watt said. "I never believed them and I was always hard on myself.
"Now he is the one that looks smart for saying that.
"I never thought I would make this impact when I came to Celtic but football is a weird game and it changes fast."
Watt will hope to continue making his mark when Celtic face third-placed Aberdeen in the Clydesdale Bank Premier League on Saturday.
"I had to pull out of the (Scotland) Under-21 squad because I was injured at the weekend and my ankle was sore," he said. "I have been getting treatment all week. Hopefully it will be okay. It's just a wee niggle."