Ashley Barnes's opener gave Burnley a shock lead heading into half time as City were left to contemplate falling to their first home defeat since December 2016, but the hosts stepped up a gear in the second half to secure their progress.
Both teams limited themselves to only four changes despite the hectic recent Premier League schedule, with Pep Guardiola retaining the likes of Aguero, David Silva, Raheem Sterling and Sane in his starting lineup.
City duly dominated possession in the first half, but clear chances were few and far between in the opening exchanges with Sane's half-volley which went out for a throw-in the most notable opportunity in the first 15 minutes.
Oleksandr Zinchenko then blasted an effort over the crossbar, but it was Burnley who had the first big chance of the contest when Barnes failed to convert Ben Mee's header back across goal from point-blank range.
Barnes was not deterred, though, and opened the scoring against the run of play after 25 minutes with an emphatic finish to beat Claudio Bravo at his near post.
A long ball upfield was flicked on by Sam Vokes before being completely missed by John Stones, allowing Barnes to capitalise on the error by racing through on goal and slamming his finish past the keeper despite the angle being against him.
Zinchenko then came close again with a curling 25-yard effort which landed on the roof of the net, but despite enjoying 74% of the ball and 12 shots in the opening 45 minutes, City went into half time without a shot on target to their name.
That would change in the second half, but only after Barnes threatened again six minutes after the restart with a long-range drive which flew over the crossbar.
Having defended stubbornly for the opening 55 minutes, Burnley's backline suffered a momentary lapse which was ruthlessly exposed by City as Ilkay Gundogan's quick free kick released Aguero, who confidently stroked his finish past Nick Pope from inside the area.
Less than two minutes later the comeback was complete as the same two players combined when Aguero exchanged passes with Gundogan, collecting the return back-heel before taking the ball around Pope and sliding it into the empty net for his 16th FA Cup goal in as many appearances.
It should have been three in the space of just seven minutes shortly after the hour mark when Pope denied Danilo from a tight angle and the rebound fell at the feet of Sane, but the German lifted his finish over the crossbar of an open goal from close range.
Sane did get his name on the scoresheet with just under 20 minutes remaining of the match, though, playing a one-two with Silva on his way into the box before holding off a challenge from Mee and burying his finish into the bottom corner.
Suddenly City were purring and looked dangerous every time they came forward, with Aguero threatening a hat-trick moments later after being found by Fernandinho inside the area.
The fourth goal did arrive eight minutes from time, though, as Kevin De Bruyne's through-ball enticed Pope off his line but was perfectly weighted for Sane, who took the ball around the stranded keeper before teeing up Bernardo to sweep his finish home just three minutes after coming on as a sub.
Man City - who will now turn their attention to an EFL Cup semi-final against Bristol City on Tuesday - are now unbeaten in their last 35 matches on English soil and 31 games at the Etihad Stadium across all competitions, while they have reached the fourth round of the FA Cup for the sixth consecutive season.
Burnley, on the other hand, are now winless in their last six matches and are forced to wait at least another 12 months to end their 104-year wait to lift the FA Cup again.
MAN CITY (4-3-3): Bravo; Danilo, Stones, Otamendi, Zinchenko (Walker, 72'); Gundogan (De Bruyne, 76'), Fernandinho, Silva; Sterling, Aguero (Bernardo, 79'), Sane
BURNLEY (4-4-1-1): Pope; Lowton, Long, Mee, Taylor; Gudmundsson (Walters, 75'), Cork (Wells, 69'), Westwood, Barnes; Hendrick; Vokes