The impressive Phil Foden hit a double after Leroy Sane made the breakthrough for City in the second half, while Padraig Amond continued his record of scoring in every round of the cup, briefly giving Newport some hope when he cut the deficit to 2-1 in the 88th minute.
More than 80 places separated the teams on the football league ladder but the plucky Welsh side made their illustrious hosts wait more than 50 minutes for a breakthrough goal.
Sane's shot was somewhat fortunate to find its way over the line having crashed into the face of goalkeeper Joe Day, while impressive teenager Foden scored a late brace to cap a fine display. Riyad Mahrez scored deep into stoppage time to give the scoreline a rather one-sided look at the end.
Newport came within inches of bringing down the house just beyond the 15-minute mark.
A long throw from Mickey Demetriou bounced all the way through to the back post and Tyreeq Bakinson stooped to head goal-ward from five-yards, only to see Ederson flying across his line to make a fine one-handed stop.
The visiting side were dominating possession and they appeared stung into life somewhat by that near miss.
Mahrez powered a shot at Day while good work from David Silva gave Sane a chance to hit the woodwork, but Newport held firm despite riding their luck.
There was certainly a hint of good fortune as they escaped the concession of a penalty with half-time in sight, Mark O'Brien diving to block Sane's goal-bound shot before the ball cannoned away off his flailing arm.
Newport survived into the dressing room but their growing hopes of an incredible upset were dealt a big blow just six minutes into the second half.
Gabriel Jesus linked with Sane down the left and German's powerful shot hit Day in the face, before looping goalwards and spinning over the line, with Foden in close attendance. The goalkeeper was left bloodied, but he continued and would make good stops from Sane, Foden and Jesus before the game was up.
Day could do nothing however when Foden picked the ball up on half-way and drove into the area before slipping a low shot underneath the arms of the goalkeeper.
Just as it seemed Newport's big night would go by without a cheer, man of the moment Amond took advantage of some defensive uncertainty from a long ball and toe-poked a shot over Ederson into the net from 20 yards out.
Their hopes were short-lived however as Foden fired a left-footed shot into the roof of the net just 71 seconds later to ensure Pep Guardiola's superstars progressed with something to spare and Mahrez added their fourth right at the death.