An own goal after seven minutes from Tim Howard was cancelled out early in the second half by Marouane Fellaini.
The Belgian then put the Toffees ahead on 72 minutes, but a late equaliser from Cottagers substitute Steve Sidwell in the 90th minute meant the Premier League match ended level at 2-2 at Craven Cottage.
Afterwards, Moyes said: "It feels like a defeat. We controlled all the game, but gave away a terrible goal at the end."
Everton had 27 shots in the match, and Moyes added: "We had total command going into the last few minutes. We should have done more with the chances we had."
Steven Pienaar returned to the side after suspension and was lively, but Nikica Jelavic's failure to turn home several good opportunities was not lost on Moyes.
"He's in a bit of a dry period at the moment," said the Everton boss.
"If he scores, we win the match comfortably. As it is, we give away a shocking goal at the end to make it two each."
Everton are up to fourth as a result of Tottenham losing at home to Wigan, and Moyes added: "The performance was as good as anything we've played this season, or last season. But it's incredible we didn't win."