The holders opened the scoring when Oleksandr Zinchenko's shot from range hit the outstretched boot of Leon Legge on its way past goalkeeper Scott Brown.
The visitors shocked Manchester City by pulling one back through Tom Pope after 35 minutes. David Amoo raced down the right and crossed for the striker to head past Claudio Bravo.
Sergio Aguero restored Manchester City's lead three minutes before half-time, tapping home Phil Foden's low cross at the far post for his 15th goal of the season.
The hosts added a third on 58 minutes through 17-year-old Taylor Harwood-Bellis, while Foden completed the scoring to complete a comfortable win for Pep Guardiola's side.
Guardiola rotated his team as expected for Saturday's clash against lower league opposition, making seven changes and naming teenagers Harwood-Bellis and Foden in his starting XI alongside experienced stars such as Aguero and David Silva.
Tyrone Marshall at the Manchester Evening News shared the two teams:
The visitors attacked the holders straight from kick-off. Scott Burgess went close with a shot that was deflected wide after a cutback from Pope.
Foden had a shot blocked by Nathan Smith and then headed straight at goalkeeper Brown at the other end before Zinchenko broke the deadlock on 20 minutes. The Ukrainian drove forwards and let fly with a rasping effort from distance that hit Legge on its way into the back of the net:
Manchester City should have doubled their lead five minutes later. Silva somehow managed to put a shot on to the crossbar from three yards out with the goal gaping, while Aguero could not convert the follow-up (UK video only):
Port Vale threatened to upset Guardiola's men with a leveller 10 minutes before half-time. Amoo produced a fine delivery from out wide on the right for Pope to beat Harwood-Bellis to the ball and head home:
The goal seemed to rouse City, and they did not take long to restore their advantage. The ball was played out to Foden on the right to cross for Aguero to tap in at the far post:
7 - Since joining Man City in 2011-12, Sergio Aguero has scored more FA Cup goals than any other player (19). Seven of these goals have come in third round matches, also more than any player has netted in a single round since he's been in England. Tremendous. #FACup https://t.co/Z6qpSSq5e1
Manchester City continued their dominance after the break and extended their lead just before the hour mark. John Stones pounced on a loose ball in the box and tucked home from close range, the ball just taking a touch off Harwood-Bellis on the goal line.
The goal was initially ruled out for disallowed but given after a lengthy VAR check:
Foden capped off another influential display with his third of the season on 72 minutes. The midfielder converted from Angelino's first-time pass after good work from Gundogan:
The result sees Manchester City begin the defence of their trophy with a comfortable win over Port Vale, who fought hard against the champions but were ultimately outclassed by superior opposition.
Manchester City play Manchester United at Old Trafford in the first leg of their Carabao Cup semi-final clash on Tuesday. Port Vale's next fixture is also on Tuesday against Salford City in the third round of the EFL Trophy.