Pep Guardiola left out John Stones and Ilkay Gundogan altogether and named Kevin De Bruyne and David Silva on the bench as City returned to action 63 hours after drawing with Borussia Monchengladbach in the Champions League.
Aguero was fit after picking up a minor foot problem and, although his goals will not win any awards, they were enough to take City back to the top of the Premier League table for a couple of hours at least.
The first goalscorer was far less predictable, Dean Marney scoring for the first time in the Premier League for seven years and in some style, but Aguero equalised eight minutes before the break and then netted the winner on the hour mark.
A calf injury picked up by Tom Heaton in training on Friday meant the end of his run of 142 consecutive appearances dating back to his debut in goal for Burnley in 2013.
The beneficiary was 37-year-old Paul Robinson, more than two years after his last senior match and four and a half years after his most recent Premier League appearance for Blackburn in May 2012.
Burnley were abject in their 4-0 defeat by West Brom on Monday but they are a different team at Turf Moor, with 13 of their 14 points coming at home.
The hosts made the brighter start and appealed for an early penalty when Jeff Hendrick went down under a challenge from Nicolas Otamendi, but referee Andre Marriner was unmoved.
City broke and Robinson was involved for the first time, getting a strong hand to Aguero's 25-yard shot.
Of all Burnley's potential sources of goals, a 25-yard volley from Marney would have been a long way down the list, but the midfielder met Otamendi's header perfectly and guided the ball into the bottom corner for his first goal since March 2014
City had been outfought by Burnley but they gradually began to improve and, after Nolito's shot had been clasped gratefully by Robinson, Yaya Toure fired against the outside of a post.
It was then City's turn to appeal for a penalty when Matthew Lowton dived in to block Nolito's shot from Raheem Sterling's cut-back, but Marriner gave a corner.
And it was from the corner that the equaliser came
Burnley failed to clear the ball and it found its way to Aguero at the back post, who squeezed his shot past Robinson.
The striker almost grabbed a second a couple of minutes later but his curling effort from outside the box was expertly tipped round the post by Robinson, who looked anything but rusty.
Burnley's afternoon took another turn for the worse when they were forced into two substitutions before the break.
Marney, who had earlier received treatment for a knock, could not continue and was replaced by Scott Arfield, and moments later Johann Gudmundsson pulled up with a hamstring problem and James Tarkowski came on.
The Clarets could have gone in ahead, though, but Sam Vokes headed Lowton's cross straight at Claudio Bravo when unmarked.
It was City who had to make a change in the 57th minute after a robust but fair tackle from Ben Mee left Raheem Sterling hobbling gingerly off the pitch to be replaced by Leroy Sane.
Three minutes later City were ahead, Aguero again in the right place to profit from some poor Burnley defending.
The Clarets looked like they had done the hard work when they stopped Toure getting his shot away but Mee and Stephen Ward tackled each other and Fernandinho's cut-back went in off Aguero's shin.
Burnley threw balls and bodies into the box as often as they could during the final half an hour but could not force the ball in, with Ashley Barnes denied twice in injury time.