Jordan Obita had given the Latics a 14th-minute lead at Boundary Park, but Victor Anichebe soon levelled and Phil Jagielka's header had Everton ahead early in the second half, seemingly for good.
But Smith came out on top in a penalty area scramble to head in an equaliser in the fourth minute of stoppage time. For the Toffees, off-colour striker Nikica Jelavic once again proved profligate on the night.
Moyes still managed to stay upbeat after the match as he said: "I'm delighted we are still in the hat for the cup."
"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."
Smith's equaliser came from the second of two Jonathan Grounds corners in a frantic finish - both of which Oldham goalkeeper Dean Bouzanis bizarrely appeared to try to punch on his foray forward.
Moyes felt his own keeper, Tim Howard, was impeded from the first set play, but praised his players for how they dealt with Oldham's approach.
"I think there is a foul on the goalkeeper when the first corner kick comes in," he added.
"I think in most games in the Premier League that would be given as a free-kick and it wasn't today.
"Most of our players are used to it (environments like this). 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."
Oldham hero Smith believes his side have "every chance" of winning at Goodison Park after his 94th-minute header earned them a fifth round replay against Everton.
"It was an unbelievable team performance from the goalkeeper right through the team," Smith told ITV.
"A fantastic performance against Premier League opposition. You can't ask for much more than that."
The entire Oldham team, including goalkeeper Dean Bouzanis, had piled into the area for the injury-time corner, and Bouzanis appeared to do enough to put off the Everton defence before Smith got the crucial touch.
"We've been working on overloading from crosses - including the goalkeeper - and it was a great cross, I just got my head on it," Smith added.
"I thought we deserved it. In the last 10 minutes we just went forward and got great efforts on goal. We were unlucky not to have scored."
Oldham, who knocked out Everton's neighbours Liverpool in the previous round, will now head to Merseyside for a replay in which Smith insists they have a strong chance.
"We have every chance," he said.
"We'll go to their place full of confidence. We've beaten Liverpool, we've got a draw with Everton, so why not?"