The Toffees were cruising deep into four minutes of stoppage time at the end of the second half thanks to goals from Kean and Dominic Calvert-Lewin but threw away the three points with some comical defending which stunned the majority of Goodison Park.
Steve Bruce's side were barely in the game – they had managed just one shot on target – when Lejeune flicked home an overhead kick.
There were possibly 90 seconds left at that point and the home side should have killed off the game, having controlled so much of it, but panicked and when the visitors pressed forward Lejeune forced home his second in a goalmouth scramble.
It ruined what should have been Kean's night after the striker finally produced the performance Goodison Park had been waiting for.
Barely a month ago Kean, memorably, was taken off by caretaker manager Duncan Ferguson just 19 minutes after coming on as a second-half substitute at Manchester United in what was arguably the lowest point of his short Everton career.
But the 19-year-old Italy international has been given a chance to impress by new boss and compatriot Carlo Ancelotti and marked his first back-to-back Premier League starts with that all-important goal in his 22nd appearance since a £27.5million move from Juventus in the summer.
Goodison had witnessed little if any of the potential talent which saw Kean score in four successive appearances last April for the Serie A champions but after six months in English football the teenager is showing signs of adapting.
Usually it is Calvert-Lewin who is Everton's all-action man up front but Kean was the Toffees' tyro in the first half.
Calvert-Lewin could satisfy himself with becoming the first Everton player into double figures in the league this season with a neat second-half finish and the first Englishman to score 10-plus for the club since since Wayne Rooney in 2017-18.
That was until Lejeune, a 70th-minute introduction for defender Ciaran Clark, produced a dramatic finish.