The League Two outfit gave the Premier League champions the briefest of scares with their 88th-minute reply before two late goals secured City their quarter-final spot.
County had previously knocked out Leicester City and Middlesbrough in the competition and did themselves justice against a side who could still win the quadruple this season.
Newport held their own for the opening half an hour and even came close to opening the scoring on 14 minutes when a long throw-in was flicked on by Jamille Matt, reaching the back post where Tyreeq Bakinson's header was pushed away by Ederson at full stretch.
Riyad Mahrez, having not started a top-flight game in 2019, seemed determined to make an impression and brought an early save out of Newport goalkeeper Joe Day, who blocked with his legs from a shot which was sneaking into the bottom corner from a tight angle.
The Man City winger tested Day again on 20 minutes after cutting back onto his left foot, while Fernandinho had a couple of his own attempts from range but the English champions were struggling to break down an opponent 82 places below them in the Football League pyramid.
When City were able to cut through the Newport defence, the crossbar came to County's rescue on 44 minutes when David Silva and Leroy Sane exchanged passes before the German lifted a shot over Day which glanced back off the crossbar and away from danger.
Sane was a growing influence as Newport tired towards the break and arguably should have won the visitors a penalty when his goalbound shot was blocked by the arm of Mark O'Brien, but Rodney Parade did not have the infrastructure to ensure that VAR was available for this tie.
Indeed it was Sane who eventually broke Newport's resolve six minutes into the second period, running onto a pass from Gabriel Jesus to unleash a shot which possessed too much power for Day to keep out, despite the County stopper getting two hands to the effort from close range.
Another perfectly-weighted pass from Jesus set up Danilo for a chance six minutes later, but this time Day got enough behind the shot to divert it onto the post.
Newport were being forced to hang on as the visitors showed their class for 20 minutes following the restart, with Guardiola still able to field a side worth in excess of £300m even after making seven changes to the team which started last weekend's 6-0 home demolition of Chelsea.
The pitch at Rodney Parade nearly proved the great equaliser for the hosts when a bobble led to Ederson scuffing a clearance on the edge of his own box, only for the Brazilian to recover before Matt could pounce.
Another opportunity to level terms passed Newport by when Mark O'Brien's header from Robbie Willmott's 73rd-minute cross was saved by Ederson.
Two minutes later, City had given themselves breathing space with a second all of Foden's making, the 18-year-old midfielder carrying the ball from the halfway line and skipping past two defenders before squeezing a shot through Day, who again may have felt he should have done better.
Guardiola started to prepare for next week's Champions League tie with Schalke 04 by taking off Fernandinho and Silva before full time, only for Newport to set up a nervy finish when Padraig Amond scored two minutes from time.
The striker capitalised on a poor defensive header from Aymeric Laporte to lift a long ball over Ederson, keeping up his record of scoring in every round of this season's competition.
However, Newport's lifeline lasted only 71 seconds as Foden swiftly restored City's two-goal lead, another fine finish from the England youth international who now has five cup goals in 12 games but is still awaiting his first Premier League start for the club.
Mahrez added a fourth in stoppage time, thrashing a shot past Day at his near post following a fine lofted pass from John Stones, adding gloss to a scoreline which perhaps flattered the Citizens given how well Newport had faired for the most part.
NEWPORT (4-4-2): Day; Butler, Demetriou, O'Brien, Poole; Bennett (Franks 86'), Bakinson (Sheehan 61'), Labadie (Dolan 68'), Willmott; Matt, Amond
MAN CITY (4-2-3-1): Ederson; Zinchenko, Stones, Otamendi, Danilo; Fernandinho (Laporte 79'), Silva (Gundogan 82'); Sane, Foden, Mahrez; Gabriel Jesus