Nathan Ake scored the only goal of the game, proving an unlikely hero as he produced a fine finish just after the hour mark.
It was close to a full strength team that Pep Guardiola fielded, with only Stefan Ortega replacing Ederson in goal as a notable change, while Mikel Arteta made six changes to the side that beat Manchester United at the weekend.
However, the Gunners competed very impressively with the Premier League champions and there was not much between the sides over the piece.
It was not exactly a cup classic, with neither goalkeeper forced into many saves and neither side at their absolute fluent best, but it was an engrossing affair nonetheless between the two best teams in the country.
City are the first team to book their spot in the fifth round, with the draw taking place on Monday 30 January.
With the strong team City fielded and the much-changed Arsenal side, many would have predicted the home side dominating the first half, but it was a very level affair.
There were not many clear cut chances at all, but it was the Gunners who had the better of them, with Tomehiro Tomiyasu and Leandro Trossard forcing saves from Stefan Ortega.
Kevin De Bruyne came closest for City as he curled one past Matt Turner, but also past his right-hand post from just outside the penalty box.
Eddie Nketiah had Arsenal’s best chance of the first period, after fine work from Trossard down the left which led to a cross which the striker flicked wide.
The second half was a bit brighter than the first, with neither side very keen for a replay clogging up their schedule and City started to impose themselves on the match a bit more.
On 64 minutes they were ahead, through an unlikely source in Nathan Ake as the centre-back produced a finish of a seasoned striker.
Substitute Julian Alvarez saw his excellent effort from range come back off the post and fall to Jack Grealish on the left of the penalty box. The England international held the ball up before finding Ake in space, who calmly curled it round the diving Turner’s left glove.
Almost immediately after the opener Nketiah came close to equalising, as Granit Xhaka whipped in an excellent cross from the left but the lunging striker couldn’t quite reach it.
Arsenal could not really press on from there though, and City were looking reasonably comfortable as the game approached the dying minutes.
There were a couple of nervy moments for Ortega in the City goal, but nothing exactly bone-shaking and the home side held on for the win.
One negative for Guardiola, though, was John Stones going off with an apparent hamstring injury during the first half.
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