As is his regular tactic in cup competitions, Leicester boss Claude Puel rotated his squad for the trip to Wales, with the likes of Jamie Vardy and Harry Maguire rested completely and James Maddison, Ricardo Pereira and Demarai Gray all left out of the starting line up.
Denis Odoi put the Cottagers ahead but Oldham battled back with goals from Sam Surridge and Callum Lang.
Premier League champions Manchester City and Watford had no such problems, claiming 7-0 and 2-0 wins over Rotherham and Woking respectively.
Will Hughes opened the scoring in the 13th minute, before Troy Deeney sealed the win with 16 minutes left in the game.
In front of their home fans at Rodney Parade, the League Two side delivered arguably the biggest upset of the FA Cup third round weekend as they saw off the 2016 Premier League winners 2-1.
The FA Cup has lost some sheen in recent years but Oldham and Barnet, along with third-tier Gillingham who beat top-flight Cardiff City on Saturday, revived the magic.
Speaking to OTB AM this morning, the Carlow man talked through his decisive spot-kick, and the satisfaction he felt at overcoming the undue attention of an English worldwide doing all he could to distract Amond at the crucial moment.
"It's Roy of the Rovers stuff". "I am the most proud man in Oldham, I can't believe it".
The victory was especially sweet for Currie, in charge after the retirement last month of John Still, as his uncle Tony was once a Sheffield United great and has a stand named after him.
Sheffield United began the day 84 places above Barnet but were outplayed by their lowly opponents for whom former Tottenham trainee Shaquile Coulthirst won it with a 21st-minute penalty after Richard Stearman brought down Ephron Mason-Clark.
Leicester supporters, however, were clearly not impressed with Puel as the 57-year-old was roundly jeered as he passed the away end and housing them to conduct his post-match media duties.