The Latics equalised deep into injury time at Boundary Park after goals from Victor Anichebe and Phil Jagielka looked to have secured the Toffees' a place in tomorrow's quarter-final draw.
The Scottish coach believes that his team should have put the game beyond doubt before Matt Smith's towering header forced a replay at Goodison Park.
"We are still in the FA Cup, but we are disappointed that we didn't win it, because we should have done better to either score another goal or to see it out," Moyes told EvertonTV.
"We had the chances to get the third goal but when you score two goals in any game you hope to win it.
"I thought we could see it out, but in the last five minutes we looked as if we were panicking a bit and we didn't do the right things."
Oldham beat Everton 1-0 at Goodison Park in the third round of the competition in 2008.