But it didn't take long for City to take the lead in the second half when Sane, one of our more unsafe players all afternoon, was played in by Gabriel Jesus before firing off a shot from a tight angle that was too hot for Day to handle.
Newport's admirable resistance was broken when Leroy Sane fired home from close range early in the second half, the winger scoring his 12th goal of the season on an afternoon when his customary flair was in short supply.
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. But Derby threatened to make a comeback after veteran left back Ashley Cole - a seven-time FA Cup victor with Arsenal and Chelsea - scored his first goal in the competition in the 81st.
Newport would have been very pleased with how they started the game, having pressed City well and dealt with their fast pace.
"Incredible", Guardiola told reporters when asked to describe the World Cup winner's performance.
"Also the fact that I've scored against the best team in Europe".
It was played on a pitch worse than the lawns in most Premier League footballers' back gardens, and there were brief moments when City came under the kind of aerial attack from set-pieces that Newport specialise in.
Newport still managed to make it interesting when Amond seized on a defensive mistake to lob the ball over Emerson, but Foden quickly restored City's two-goal lead with a fierce left-foot shot into the net from inside the area. I am surprised they are in the position they are. As the pitch disintegrated quicker than a digestive biscuit, so City's great players adapted to its unpredictability and their mastery of the ball, even in these conditions, was spectacular at times. "The support for the team was incredible, chanting: 'Who are you, who are you?'" Now we go to biggest stage, Champions League, and we will see in May and June.
While now 15th in the League Two table, Newport County made the deepest FA Cup run of any fourth-tier side in this season's competition. "I'm trying to take my opportunities as they come".
"Also the backroom staff, I can sort them out as well".
"I am a little bit disappointed with the last two goals we conceded, because we got back into it and we just wanted that one chance, but we didn't hold on".
"I was buzzing at half time, especially after watching them [against Chelsea] last week".
"We are still in all competitions and we want to keep fighting for every one - and hopefully we can win them all".
"Getting to share the touchline with Pep Guardiola was magical for me". He said "you should be proud, you are doing a fabulous job, and good luck".
There will be a considerable number of changes, but City's squad depth means we are likely to see another formidable Manchester City XI take the field in a few hours. The players are full of class, the manager is full of class. "He is a serial victor and he is also full of respect".