The League One side needed a last-minute goal from Matt Smith to level the scores, and secure a replay against the Toffees at Goodison Park.
The Latics caretaker boss told BBC Sport: "Everyone near me was jumping around when Matt scored but I just started smiling to myself. These have been troubled times and I am under no illusions what a game at Goodison Park means to us financially.
"From the club's perspective, drawing with Everton is massive. For me personally, seeing the development of my young players earlier today meant just as much. I just said to the players 'enjoy tonight' but our game at Stevenage on Tuesday night is as important."
The cash-strapped side had to ban players from swapping shirts with their Premier League opponents at the end of yesterday's tie due to a shortage of first-team shirts.
Oldham will be in the draw for the last 16 of the FA Cup, which takes place later, for the first time since 1994.