Aguero's decisive brace took his remarkable scoring run to 33 goals in 34 Premier League games after Dean Marney had given Burnley a surprise lead over the title contenders.
City's recovery secured their sixth win in seven away games and moved them to the top of the table, although they could surrender pole position to Liverpool or Chelsea later in the day.
It was the first time this season that Pep Guardiola's team had come from behind to win, and the first time since September they had won back-to-back league games, having triumphed at Crystal Palace last time out.
Burnley sprung a surprise by including former England goalkeeper Paul Robinson, now 37, after Tom Heaton injured a calf during training to miss his first game since joining the club.
Robinson, signed on a free transfer from Blackburn, had not played in the Premier League since 2012 and had gone 802 days without a first-team game, but he showed he still has the ability that earned him 41 caps and a career that took in Tottenham and Leeds.
Burnley were defending one of the best home records in the Premier League, having won four of their previous six games - a run bettered only by Chelsea, Liverpool and Tottenham - and they took the game to City.
The visitors were fortunate not to concede a penalty as early as the fifth minute, but the referee Andre Marriner waved away the protests when Jeff Hendrick went down after an elbow in the back from Nicolas Otamendi.
Guardiola made five changes from the team that played in midweek in the Champions League, but they struggled to come to terms with Burnley's approach and energy.
Robinson did have to tip an effort from Argentina forward Aguero round a post but then, on 13 minutes, the goalkeeper initiated the opening goal as Burnley took the lead.
Robinson launched a long free-kick into the City penalty area and Otamendi's attempted clearing header fell invitingly for long-serving midfielder Marney, who rifled the ball past Claudio Bravo's dive for his first Premier League goal in seven years.
City responded but Robinson made a fine one-handed save to deny Nolito and Aguero lifted his own effort over the bar from Fernandinho's lay-off.
Burnley eventually succumbed on 36 minutes and almost inevitably, it was Aguero who scored his ninth goal in 11 Premier League games and 15th in all competitions, pouncing at the far post after Nolito's initial shot from a corner had been blocked.
The scene appeared set for City to stride into the lead before half-time, but Robinson produced his best moment so far, a superb one-handed flying save to keep out Aguero's fierce shot on the turn.
Burnley, having lost the advantage, also lost scorer Marney to an injury and then Iceland midfielder Johann Gudmundsson limped off with a hamstring strain before half-time.
City took the lead on the hour after Yaya Toure went down in the area and Burnley made a hash of clearing before Fernandinho was able to cut the ball back towards the near post where Aguero put the ball in with his knee.
Burnley refused to go down without a fight, but Michael Keane's well-directed header was cleared, George Boyd had a shot blocked, and substitute Ashley Barnes had an overhead kick well saved as City held out.