The Premier League side looked to be heading through to the quarter-finals as goals from Victor Anichebe and Phil Jagielka put them in front after Jordan Obita had fired Oldham ahead early on.
But substitute Matt Smith headed home a corner in a packed penalty box to secure a 2-2 draw and a replay at Goodison Park.
Moyes was gracious enough to admit that the League One side were good value for a share of the spoils, but felt goalkeeper Tim Howard was impeded in the dramatic closing stages.
The Scot told Sky Sports: "It is a tough place to come and Oldham made it difficult. I think they deserved to get the replay.
"But we had great chances to wrap it up when it was 2-1. We missed a couple of big chances to make it 3-1.
"I think there is a foul on the goalkeeper on the first corner that comes in.
"In most other games in the Premier League that would have been given as a free-kick and it wasn't.
"Most of the players have been brought up through this level and we warned them and told them - we have played quite a few games like this, and we have lost quite a few games like this.
"So there has to be a bit of credit for how we stuck at it and did the job.
"Other teams have come and it has not happened that way. We did and we'll have a second chance to try to get through."