Having knocked out Liverpool in the previous round, Oldham again had Merseyside opposition reeling when on-loan Reading winger Jordan Obita gave them a 13th-minute lead.
Victor Anichebe ensured Everton were level midway through the first half before they took what appeared to be a decisive lead shortly after the interval, Phil Jagielka heading home Kevin Mirallas' corner.
Off-colour striker Nikica Jelavic once again proved profligate and Matt Smith had the final word deep into injury time as the League One club salvaged a 2-2 draw and a replay.
"I'm delighted we are still in the hat for the cup," Moyes said.
"It is a tough place to come - Oldham made it difficult and I think they deserved to get their replay.
"But then I also think 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 and we didn't do so.
"But hey, that is what the cup does. Give Oldham credit - they have taken some big scalps in this competition already and they nearly did it again."