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.
But, with five minutes left, Newport were awarded a penalty after Leicester's Marc Albrighton inexplicably raised his arm to block a cross into the box and Padraig Amond made no mistake from the spot to spark delirious scenes among the Welsh club's supporters.
Newport County striker Padraig Amond celebrates after scoring their 2nd goal from the penalty spot during the FA Cup match against Leicester City at Rodney Parade in Newport. He faced a few of my penalties in training so I was hoping he wouldn't remember.
There was no such misfortune for Manchester City as they cruised to a comprehensive 7-0 victory over Rotherham United, matching Tottenham's tally against Tranmere Rovers, with Watford also comfortable winners at Woking.More news: During Shutdown, National Parks Are Not Doing Well
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"I'm so proud of the players", Newport manager Michael Fylnn told BBC Wales.
Claudio Ranieri's Fulham reign has got off to a hard start and it got worse for the former Leicester boss at the hands of Pete Wild's Oldham.
Leicester have never won the FA Cup having been runners-up four times, the last in 1969.
Flynn added: "It's nearly Roy of the Rovers stuff, I can't believe it". Barnet is now managed by Darren Currie, the nephew of Sheffield United great Tony Currie. There were one or two scary moments but we came through it.
City boss Pep Guardiola named a strong team with the starting eleven featuring the likes of Ederson, Kyle Walker, John Stones, Kevin De Bruyne, Riyad Mahrez and Raheem Sterling. Championship strugglers Rotherham offered some resistance, but any hopes of a giant-killing were ended by half-time - Phil Foden and a Semi Ajayi own goal putting the result beyond doubt.