Kean came on as a second-half substitute while Juve was being held to a goalless draw and scored his third goal in as many Serie A appearances just three minutes after his introduction.
The 18-year-old has hit form at an opportune moment for Juventus, which is without the talismanic Cristiano Ronaldo because of a hamstring injury, but Allegri warned that many promising youngsters have "lost their way over the years".
"It's only right to have ambition when you are training with Juventus and surrounded by these champions," Allegri said.
"The difference at this level is mentality and he [Kean] needs to above all be professional over time, be ready to work hard and improve every day.
"He's got the characteristics and the quality to be a great player, but so did many lads who lost their way over the years."
Paulo Dybala also missed the win over Empoli after pulling up with a thigh injury during the warm-up and Allegri admitted that it took his side a while to get going in the absence of key players.
The visitors belied their lowly league position of 17th and had the better of the first half but Allegri said his side's patience paid off.
"Dybala had a knock during international duty and he felt the muscle harden during the warm-up, so we decided not to risk it," the manager said.
"We had prepared Dybala between the lines, but losing him at the last minute I had to make some changes. I decided to wait until half-time to make the necessary adjustments, as it was quite difficult to get the message across.
"Empoli [was] very aggressive early on and we showed patience, waited for them to drop and raised our tempo, revving up the engine. Sometimes you need to do that; it's not that because we are Juventus we automatically win every game.
"We struggled in the first half and couldn't get a grip on Empoli's shape, but we changed some things and had an excellent second half, not allowing them a shot on goal."
Looking ahead to his side's forthcoming fixtures, which includes Serie A games against AC Milan and Inter Milan and a UEFA Champions League quarter-final against Ajax, Allegri said: "Now we have seven games in 20 days and fortunately we are recovering some players who can give us a big hand."