Manchester City recorded their biggest FA Cup victory for 51 years, thrashing Championship side Rotherham United 7-0, but it was a day for the lesser lights, with bookmaker William Hill quoting odds of over 450,000-1 for Newport, Oldham and Barnet all to win.
Puel was sharply criticised for naming an under-strength side for the Carabao Cup quarter-final defeat by makeshift Manchester City opponents last month.
Sam Surridge equalised with a penalty but Fulham missed a spot kick through Aleksandar Mitrovic, who was brought on as a substitute to take the kick, before Callum Lang snatched the victor two minutes from time.
"It's unbelievable", Barnet captain Callum Reynolds said after the north London club's triumph.
"I thought we played well, especially the second half".
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"I mean they (Sheffield United) are third in the Championship and we beat them at their ground".
QPR defeated Championship leaders Leeds by the same scoreline, Jake Bidwell grabbing a 75th-minute victor for the west London club.
Manchester City joined Manchester United, Tottenham Hotspur, Chelsea and Arsenal in the fourth round as they thrashed second-tier Rotherham United 7-0.
An inexperienced Leeds side, meanwhile, came unstuck at Loftus Road, where Jake Bidwell's header ended QPR's long wait for a win in an FA Cup tie without having to go to a replay.
Elsewhere, in an all-Premier League clash, Brighton won 3-1 away to south coast rivals Bournemouth.
Sheffield United, third in the Championship and vying for automatic promotion to the Premier League, could not find a way back into the game as Chris Wilder's men suffered a humbling defeat at the hands of the non league side, 15th in the National League.