The home side took the lead in the seventh minute when Bryan Ruiz’s free-kick struck the post and rebounded off Tim Howard and into the Everton net.
However, Everton midfielder Marouane Fellaini scored a second-half brace to leave the Toffees heading for maximum points in west London.
But Fulham, who played out a dramatic 3-3 draw with Reading last weekend, equalised in the 89th minute when Steve Sidwell drifted past Seamus Coleman to finish from close range.
“It feels like a defeat,” Moyes told BBC Sport. “We conceded a shocking goal to allow them to equalise at the end. We’re as sick as it gets.”
Martin Jol admitted the Cottagers were fortunate to equalise and the Fulham boss was satisfied with a point after a disappointing performance.
“You need a little bit of luck and we certainly got that,” said Jol.
“Everton could have scored a third goal and a fourth goal, but we never gave up. To come away from a bad day like today with a point is great.
“If you saw us play in this game, we’d be happy to finish in the first 10 this season.”
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