The Argentina striker netted his 177th City goal from a controversially-awarded first-half penalty to set City on their way to a club record-equalling 11th successive win in all competitions.
Burnley proved tough opponents amid torrential rain for Pep Guardiola's free-flowing side but late goals from Nicolas Otamendi and Leroy Sane secured another three points.
Aguero's goal took him level with Eric Brook, who scored 177 for City between 1927 and 1939, but he will have to wait to claim the record outright after being substituted in the 76th minute.
It was his first appearance since breaking a rib in a car crash three weeks ago.
He actually had a quiet start to the game as City, despite dominating possession, struggled to break Burnley down.
The Clarets created the first serious opportunity when Chris Wood broke clear of Kyle Walker and raced into the box.
Ederson dived at his feet and quickly rose to prevent Scott Arfield following up with a shot but Wood was injured in the collision and was forced off soon after.
City played a patient game and gradually began to create chances. Bernardo Silva should have opened the scoring after Sane skipped around a challenge and pulled the ball back but the Portuguese shot straight at Nick Pope.
Controversy erupted after Pope reacted quickly to deny Kevin De Bruyne the chance to shoot. The ball ran free and Pope was adjudged to have brought down Bernardo Silva in the rush to regain possession.
Burnley contested that Silva had made the most of any contact and there was some arguing and shoving amongst players.
Aguero remained calm throughout the confusion and, having missed from the spot on his last appearance, made no mistake to put City ahead and equal the record.
The perceived injustice did spur Burnley on and Arfield had a shot deflected wide.
But City also reacted strongly. Aguero blasted wide and had two good efforts saved by Pope. The ball ran free to Bernardo Silva from one of them but James Tarkowski, not entirely intentionally, got his head in the way of a powerful shot.
City continued to dominate after the break, although Burnley retained their shape and did cause a few moments of anxiety.
David Silva threatened with a long-range drive that screwed wide while Walker delivered a low ball across the box that Aguero could not turn in.
The second goal eventually came after 73 minutes and from an unconventional route - for City at least - as Otamendi got forward for a corner. The Argentinian met Sane's cross with a powerful header that Steven Defour could not keep out.
Moments later the points were wrapped up as De Bruyne picked out Sane with a now typical defence-splitting through-ball. The German raced into the area and lashed a low shot past Pope.
Substitute Gabriel Jesus and Sane went close to adding to the score in the closing moments.