The Iceland midfielder struck in the 77th minute, skilfully turning away from James Maddison and bending an effort from 25 yards into the top corner.
Just 14 minutes earlier, Foxes captain Morgan received his second booking in 20 minutes for a foul on Richarlison.
Richarlison had put Everton ahead on seven minutes with a close-range volley, set up by some wonderful trickery by fellow Brazilian Bernard, who was making his first Premier League start since his summer move from Shakhtar Donetsk.
Just before half-time Ricardo Pereira scored a deserved equaliser for Leicester which came from an Everton corner.
The Portuguese full-back, led a counter-attack exchanging passes with Kelechi Iheanacho and then left Toffees defender Jonjoe Kenny on the floor as he twisted and turned inside the box before firing in at Jordan Pickford's near post, the final touch coming off the England keeper.
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