Frank Lampard's newly-inherited side went ahead in the tenth minute after a cross from Donny van de Beek was headed home by Seamus Coleman from close range.
Leeds had lost Stuart Dallas to injury one minute earlier and United's afternoon took another turn for the worse when Everton doubled their advantage in the 23rd minute as Michael Keane headed home a corner.
Rodrigo had been denied by the crossbar after a fine volley moments earlier but the Toffees created a host of chances, substitute Leo Hjelde producing a fine tackle to deny Richarlison and Illan Meslier saving well to push a Dominic Calvert-Lewin shot wide.
The half then ended with Rodrigo being denied by the crossbar for a second time and in similar circumstances, a rising shot from the edge of the box smacking back off the frame of the goal.
Whites head coach Marcelo Bielsa opted to make a double change during the interval and it was star man Raphinha and Mateusz Klich who were taken off as Tyler Roberts and Adam Forshaw were introduced.
Roberts went up front but the only real chance fell to Dan James who cut in and fired wide and Everton added a third in the 78th minute when Richarlison danced around the box before firing a low shot past Meslier into the bottom left corner.
The Toffees then would have added a fourth in the 90th minute but for a superb save from Meslier to tip a fierce Salomon Rondon volley over the bar and United then lost Hjelde to injury deep in second half stoppage time.
Graham Smyth, the YEP's chief football writer, hands out the scores.

7. Raphinha
5 - Scared Everton when he got on the ball but defensively looked very suspect. Withdrawn at the break. Photo: Getty Images

8. Mateusz Klich
4 - Couldn't control the midfield on his own, looked isolated and did little to stand out. Photo: Getty Images

9. Rodrigo
5 - Hit the bar twice with audacious efforts, played a few nice passes but midfield looked vacant when Everton had the ball. Faded. Photo: Getty Images

10. Jack Harrison
5 - Some nice footwork aside he was unable to get the better of his markers. Coleman gave him little joy, either did Kenny. Photo: Getty Images

11. Daniel James
4 - Looked isolated up front at times as long balls did little for him. When he got on the ball he was unable to do much with it. Photo: Getty Images

12. Adam Forshaw (sub)
6 - At the very least he helped Leeds retain possession a bit more in the second half. Leeds still looked vulnerable out of possession. Photo: Getty Images

