David Moyes was left to rue the panic caused by Oldham's late onslaught.
After Everton had recovered from going a goal down to lead 2-1, they conceded a last ditch equaliser as Oldham forced a replay.
First Victor Anichebe levelled before Phil Jagielka glanced the Blues ahead. But Matt Smith struck with the last act of the game to make it 2-2 after the home side had poured on the pressure.
Moyes told evertontv: "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.
"We had the chances to get the third goal but when you score two goals in any game you hope to win it."
Oldham's late charge saw them launch the ball high into Everton's box time and again, before forcing a succession of late corners.
"We just couldn't get a head or the goalie's hands on it," said Moyes of the final set-piece. "I thought there was a foul on the goalkeeper from the previous corner.
"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."
The replay takes place at Goodison Park on 27 February, at 7.45pm.