City, third in the Premier League table, fell behind after seven minutes when Harry Bunn drilled an effort beneath keeper Claudio Bravo but that served only to stir the hosts into action.
Leroy Sane drew them level before a penalty from Aguero and a tap-in from Pablo Zabaleta gave them a 3-1 advantage at the break, with their first three goals coming in a devastating eight-minute spell.
Aguero grabbed a fourth in the second half and Kelechi Iheanacho completed the rout in stoppage time as City secured an away tie against Premier League Middlesbrough in the last eight.
After a goalless first game, the deadlock was broken early as Huddersfield stunned the Etihad Stadium when Bunn found the net.
Having been teed up by Philip Billing, his low effort squeezed beneath City keeper Bravo, who has been demoted to second choice after a series of unconvincing displays.
Despite the early setback, City began to dominate and were rewarded for their pressure when Sane finished from close range after 30 minutes.
The goal turned the momentum firmly in City's favour and they took the lead four minutes later as Aguero was bundled to the floor and stepped up to slam his penalty high into the top corner.
City's recovery was completed with a third goal in quick succession as Aguero picked out Zabaleta for a simple finish to effectively wrap up the tie.
They added to their advantage with 17 minutes remaining as Raheem Sterling's cross was expertly turned home by Aguero at the near post and Iheanacho tapped home a low cross at the death.
(Reporting by Toby Davis, editing by Ed Osmond)
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