Earlier, Fulham were up-ended 2-1 by fourth-tier side Oldham.
Jamille Matt headed the Exiles in front but Rachid Ghezzai levelled for the visitors on 82.
However, moments later a penalty was awarded to Newport when the ball struck Marc Albrighton's arm, which was converted by Padraig Amond in the 85th minute.
"I'm so proud of the players", Newport manager Michael Fylnn told BBC Wales.
"I thought we played well, especially the second half". We had eight players who won the title.
Denis Odoi put Fulham 1-0 up after 52 minutes but Oldham hit back through Sam Surridge's penalty before Aleksandar Mitrovic, on as a substitute, had a spot kick of his own saved.
Substitute striker Aleksandar Mitrovic then missed a penalty kick, before Oldham's Callum Lang headed in the victor in the 88th minute.
Remarkably, both Oldham and Barnet have caretaker managers - Pete Wild and Darren Currie - who were taking charge of their respective teams for only the third time. "The character in the dressing room is phenomenal".More news: Kevin Spacey to appear in court on sexual assault charge
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Puel (above) also left out Vardy and made changes to his starting line-up when Leicester were knocked out of the Carabao Cup in the quarter-finals, beaten in a penalty shoot-out by Manchester City.
This edition marks the first time Barnet have made the fourth round since 2008 when they lost to Bristol Rovers, and will now have their sights firmly set on reaching the last 16 for the first time ever.
Foden's goal was the England Under-21 international's first in senior football at the Etihad.
"It's unbelievable", Barnet captain Callum Reynolds said after the north London club's triumph.
But hopes of Woking, 110 places adrift of the Hornets, enjoying a shock win suffered a setback when Will Hughes curled in a 13th-minute shot from the edge of the area.
Watford doubled their lead 16 minutes from time through Troy Deeney.
Manchester United are on their longest winning run in 12 months, with their win over Reading making it five on the bounce under Ole Gunnar Solskjaer.