The match appeared to be meandering towards a stalemate until James Tarkowski was harshly adjudged to have handled Anton Stach’s driven shot.
Nmecha had only been on the field for six minutes of his Leeds debut but confidently fired past Jordan Pickford in the 84th minute.
“It’s not a penalty,” said Tarkowski. “I was pretty convinced it was going to get overturned (by VAR). A bizarre moment and it cost us the point in the end.”
Everton handed a debut to Manchester City loanee Jack Grealish in the second half, but the Toffees were left to rue not posing anything of an attacking threat in the opening 45 minutes.
Leeds are aiming to avoid the trend of the past two seasons when all three promoted clubs have been relegated.
Daniel Farke has also struggled previously with the step up to the English top-flight during two attempts as Norwich manager.
But after Sunderland beat West Ham 3-0 on Saturday, two of the three promoted sides have won on the opening weekend for only the third time in the Premier League era.
Farke’s men stormed to promotion by winning the Championship title with 100 points last season and Leeds started in confident fashion.
The home side were dominant in the opening half with 12 shots on goal but only once made Pickford work when the England goalkeeper blocked from Joel Piroe.
Grealish’s introduction coincided with Everton’s best spell of the game.
Lucas Perri, one of Leeds’ eight summer signings, made an impressive save at his near post to turn behind Carlos Alcaraz’s powerful strike before Jake O’Brien headed the resulting corner over.
Just as Leeds looked like they might have to hold on for a point, they were gifted the chance for a perfect start with all three.
There was little Tarkowski could do as his left arm was down by his side as it blocked Stach’s effort.
Nmecha stayed calm amid the furore as the German international found the bottom corner.
“For a newly-promoted side it (Elland Road) has to become a fortress,” said Farke.
“It’s great to see the joy and the pride in their eyes. It’s great to give the supporters, who are so emotional and care so much about their club, something to cheer,” he added.
Tarkowski had Everton’s only chance to make amends with an equaliser but he headed over. (AFP)