The visitors started brightly in west London and could have been ahead inside eight minutes when Jake O’Brien headed on to the post from James Garner’s corner.
Fulham, though, hit back with a chance from Alex Iwobi – who spent four years with the Toffees prior to joining Fulham – before Mykolenko’s unfortunate own goal put the home side ahead in the 18th minute.
It came after a slick move from Marco Silva’s side – started by the excellent Samuel Chukwueze – and although Jordan Pickford saved well from Raul Jimenez, the ball ricocheted off the Ukraine left-back and into the back of the net.
If there was a slice of luck about Fulham’s opener, a one-goal advantage was the least they deserved as they dominated the rest of the first half. Pickford was called into action again as he produced a stunning fingertip save to turn Chukwueze’s effort around the post shortly afterwards.
Fulham boss Silva was left wondering how his side did not add to their lead as Emile Smith Rowe and Chukwueze rattled the crossbar from long range in the final 10 minutes of the half.
In a more evenly matched second half, Everton manager David Moyes – serving a touchline ban – sent on Beto and new signing Tyrique George in search of an equaliser with 20 minutes remaining.
They found it when Mykolenko made amends for his earlier mistake by breaking away from Timothy Castagne to tee up Dewsbury-Hall to convert from close range.
The travelling Blues were still celebrating that moment when Leno diverted the midfielder’s corner into his own goal seven minutes later to hand Moyes’ side a dramatic victory as their impressive away form continued.