Oleksandr Zinchenko, Sergio Aguero, Taylor Harwood-Bellis and Phil Foden were on target for Man City but Tom Pope had a moment to savour.
Port Vale's Tom Pope had the FA Cup impact he promised before holders Manchester City ran out 4-1 winners at the Etihad Stadium.
Pope, whose disparaging Twitter post about City defender John Stones came to light following the third-round draw, nodded in a superb 35th-minute equaliser after Oleksandr Zinchenko's deflected opener.
However, Sergio Aguero ensured the hosts went in ahead at half-time before youngsters Taylor Harwood-Bellis – inadvertently scoring his first senior goal after Stones thought he had City's third – and Phil Foden sealed the Premier League champions' progress.
Nathan Smith scrambled an eighth-minute effort from Foden out of the goalmouth but fellow Vale centre-back Leon Legge was not so effective with his own last-ditch attempt – turning Zinchenko's speculative 20th-minute strike past Scott Brown.
David Silva inexplicably scuffed Ilkay Gundogan's low cross into the turf and against the crossbar with the goal at his mercy and City would rue that miss.
On a rare Vale attack, David Amoo raided down the right and Pope found space off the back of Harwood-Bellis to dispatch a fine header.
Another player who would certainly fall under Pope's self-applied label of "bagsman" is Aguero, and City's all-time record goalscorer restored his team's lead after Foden stayed just onside to steer Gundogan's chipped pass across goal.
Brown made superb early second-half saves to deny Bernardo Silva and Foden, yet City had the breathing space they desired before the hour.
Stones looked to have levelled the personal duel with Pope – insofar as there was one – but a VAR check showed his close-range shot clipped Harwood-Bellis on the line.
A wretched clearance from Stones, half-blocked by Pope, meant Vale substitute Tom Conlon almost reduced the deficit with a diving header, while David Silva's frustrating outing in front of goal continued as he dragged wide.
But Foden got the goal his efforts deserved from a drilled Angelino cross, allowing attentions to turn towards a blockbuster midweek derby.
What does it mean? Guardiola's Cup kings march on
Pep Guardiola selected a team with one eye on the forthcoming first leg of the EFL Cup semi-final against Manchester United and City were far from their fluent best. Still, City have now prevailed in 22 of their past 23 domestic cup ties, with a 2018 FA Cup fifth-round defeat to Wigan Athletic the only blot in the past three seasons.
Pope walks the walk
Vale's veteran striker arguably set himself up for a fall seven months before this tie with a tweet claiming he would "get 40 a season" playing against Stones. Pope's bravado was duly unearthed after the draw and, although the visitors' goal had nothing to do with any short-comings on Stones' part, he relished backing up his bold words.
City understudies come up short
Angelino and Claudio Bravo should probably thank Pope for targeting Stones on social media, because their parts in the goal might be overlooked. The left-back's lacklustre efforts when it came to stopping Amoo's cross left City in trouble. It was a fine finish but, not for the first time in his City career, Bravo appeared awfully flat footed, meaning his chances of pulling off the save were limited to begin with.
City resume their other domestic cup defence with the first of a Manchester derby double-header at Old Trafford on Tuesday, while there is a Manchester United-flavour to Vale's midweek assignment as they travel to Salford City in the EFL Trophy on the same day.