The upset looked to have been averted but for Marc Albrighton to raise an arm to block a cross to give away a penalty that Padraig Amond slotted away superbly with five minutes left to hang on for a famous win.
Premier League champions Manchester City brushed aside Rotherham 7-0 and Watford secured a 2-0 win away to Woking, but Fulham threw away the lead in an embarrassing home defeat to League Two opposition.
Puel saw his side beat Chelsea, Manchester City and Everton - while losing to Cardiff City - over the festive period, and his decision to make seven changes backfired when Jamille Matt marked an outstanding display by heading home in the 10th minute from Robbie Willmott's pinpoint cross.
City added to their lead in the second half with Gabriel Jesus tapping in from Sterling's cross and Riyad Mahrez firing in after playing a one-two with Ilkay Gundogan.
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Callum Lang then sent 4,000 travelling Oldham fans wild with a headed victor in the 88th minute.
Padraig Amond was the match victor for the Exiles, scoring a late penalty not long after Rachid Ghezzai had equalised for the Foxes.
Flynn added: "It's nearly Roy of the Rovers stuff, I can't believe it".
Barnet also prevailed at Championship high-flyers Sheffield United, where Shaquile Coulthirst's first-half penalty making it a sweet victory for Bees' caretaker manager, Darren Currie. I'm so proud of the players'. Barnet is now managed by Darren Currie, the nephew of Sheffield United great Tony Currie.
"I asked them to play with no fear today and I think they managed to achieve that".