Padraig Amond was the match victor for the Exiles, scoring a late penalty not long after Rachid Ghezzai had equalised for the Foxes.
After being criticised for fielding a weakened side in the Foxes' Carabao Cup quarter-final defeat to Manchester City last month, Puel's much-changed line-up on Sunday drew added scrutiny.
"It is Roy of the Rovers stuff", said Oldham boss Pete Wild.
Defender Denis Odoi gave Fulham the lead shortly after halftime, before loan striker Sam Surridge levelled from the spot in the 76th minute.
Striker Shaquile Coulthirst scored a 21st-minute penalty to send Barnet through to the fourth round, where they'll be the only non-league team to feature in the round of 32.
However, Foxes captain Morgan - who started the match - said: "I think any criticism (of Puel) would be unfair".
Oldham came from behind to win 2-1 at Fulham, a new low for the team managed by Claudio Ranieri which is fighting relegation in the Premier League.
"In France it is also a big tradition and we know from all the teams playing in different leagues it is a fantastic opportunity to play against a Premier League team and to try to create an upset".
QPR defeated Championship leaders Leeds by the same scoreline, Jake Bidwell grabbing a 75th-minute victor for the west London club.
While all of the Premier League's big boys made it through to the fourth round unscathed, there were a few casualties from the top flight, with Leicester's defeat to Newport and Fulham's defeat to Oldham being the standout results.
Sheffield United may be riding high in the Championship but their FA Cup campaign ended at the first hurdle as Vanarama National League side Barnet won 1-0 at Bramall Lane.
Elsewhere, in an all-Premier League clash, Brighton won 3-1 away to south coast rivals Bournemouth.
Manchester City, meanwhile, thrashed Rotherham 7-0 at the Etihad as Pep Guardiola named a strong side following the win over Liverpool on Thursday.
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