After a 0-0 draw in the original tie, the visitors led when Harry Bunn's shot went through Claudio Bravo's legs.
But tap-ins from Leroy Sane and Pablo Zabaleta, with Sergio Aguero's clinical penalty in between, turned the replay in City's favour before half-time.
Aguero swept in Raheem Sterling's cross at the near post for City's fourth, before substitute Kelechi Iheanacho poked in with the last kick of the game.
City's reward is a quarter-final trip to Premier League rivals Middlesbrough on Saturday, 11 March (12:15 GMT).
Pep Guardiola arrived at Manchester City last summer with the clear remit from owner Sheikh Mansour to take the club to "a new level".
While that demand is largely thought to mean success in the Champions League, the City hierarchy will also want to see the former Barcelona and Bayern Munich coach deliver domestic trophies.
Winning the Premier League looks to be a uphill task with Chelsea 11 points clear, while Manchester United took the season's first silverware by claiming the EFL Cup.
But they have moved into the last eight of the FA Cup for the first time in four seasons after overcoming an early scare against Huddersfield.
"Seeing their biggest rivals United bag the first trophy of the season means City will be thinking 'we want a piece of that'," said Match of the Day pundit Alan Shearer.
"I think this is their best chance of silverware this season."
Despite the relative ease of extending their unbeaten run to eight matches, there were still some concerning sights for Guardiola. Namely more uncertain goalkeeping from the returning Bravo.
Guardiola reiterated his confidence in the former Barcelona keeper before the replay, despite trusting Willy Caballero instead in City's past four Premier League matches and the Champions League last-16 win against Monaco.
However, he watched the Chilean make another error to gift the opening goal to former City youngster Bunn.
The 33-year-old stopper, who was dropped after conceding 16 goals from the previous 24 shots on target he had faced in the Premier League, let Bunn's low strike through his legs in Huddersfield's first effort on target.
That led to ironic applause from the home fans when Bravo blocked Huddersfield's second shot at goal shortly before half-time.
Guardiola did not appear pleased, turning around to glare at the supporters behind him.
Afterwards the Spaniard described Bravo's performance as "top", adding "his performance with the foot helped us build up".