Phil Foden opened the scoring just seven minutes into the tie, before adding a second in the 74th minute with a deflected effort, which still may go down as an own goal from Zak Vyner.
Kevin De Bruyne then sealed the victory with a sweet strike in the final 10 minutes to fire Pep Guardiola's side into the quarter-finals for the fifth successive year.
Man City have now won each of their last 11 away matches in the FA Cup, the longest winning run on the road by any team in the competition's history.
The Citizens have also extended their winning run against lower-league teams in the FA Cup to 13 games, while Bristol City have now been eliminated from 14 of their 15 FA Cup fifth-round ties.
Kalvin Phillips was one of six changes made to Guardiola's starting lineup following last weekend's 4-1 Premier League win at Bournemouth, with the midfielder starting just his second game of the season since joining the Citizens last summer.
The Yorkshireman was instantly involved in the action at Ashton Gate and almost opened the scoring in just the second minute when he rattled the crossbar with curling shot from around 20 yards out.
However, Man City made the breakthrough just five minutes later when a neat one-two between Riyad Mahrez and De Bruyne resulted in the former fizzing an inviting delivery towards the back post for Foden to tap home a first-time finish from a tight angle.
As expected, the visitors dominated proceedings but Nigel Pearson's side had their fare share of chances in front of the fired-up Ashton Gate faithful, with highly-rated duo Alex Scott and Sam Bell posing a constant threat in and around Man City's box.
The Citizens ended the first half strongly, with De Bruyne and Ruben Dias both coming close to doubling their lead, but the Robins defended well to ensure they headed into the interval only a goal down.
Both teams made a change at the break, with Man City goalkeeper Ederson replacing Stefan Ortega who required treatment for a hand injury early in the first half, while Bristol City opted for an attacking alteration, with skipper Andreas Weimann introduced at the expense of George Tanner, with Mark Sykes filling in at right-back.
Ederson was instantly called into action and was involved in a nervy moment on the edge of his box, clashing shins with Bell after clearing the ball down the pitch.
Just a few minutes later, Foden - who started the second half as brightly as the first - drove down the left flank and almost picked out Julian Alvarez with a dangerous cross across the six-yard box.
After Alvarez tested Max O'Leary from range, Bristol City enjoyed a positive spell of possession in the final third and arguably their best chance of the match fell to Bell inside the six-yard box on the 70-minute mark.
Dias looked to shepherd the ball out for a goal kick, but Scott's relentless efforts to win the ball back paid off, flicking the ball back for Weimann to whip a dangerous delivery towards Bell, who directed his first-time effort off target.
Just as Bristol City appeared to be getting back into the contest, Man City doubled their lead in the 74th minute when Alvarez teed up Foden for a left-footed strike inside the area, which took a deflection off Robins defender Vyner before finding the bottom corner.
De Bruyne then put the game to bed seven minutes later with a perfectly-placed curling strike from around 25 yards out into the bottom-right corner.
Man City remain on course to lift their first FA Cup since 2018-19, and their name will be in the hat for the quarter-final draw on Wednesday night.
Guardiola's side will now turn their attention to Saturday's Premier League clash at home against Newcastle United, while Bristol City will look to return to winning ways when they travel to South Wales for the Severnside derby against Cardiff City in the Championship.