Aguero’s first-half goal, from a controversial penalty, was his 177th for the club and brought him level with Eric Brook’s record, as well as sending City on their way to an 11th straight win, equalling a club record.
But they had to work hard to see off a resolute Clarets side, who were unhappy with the decision that led to them going behind.
A pack of Burnley players protested when Roger East pointed to the spot after visiting keeper Nick Pope tangled with Bernardo Silva, but replays showed Pope’s knee came down on Silva’s foot as he slid in.
Aguero stepped up to calmly slot home, but Burnley, who arrived at the Etihad on the back of a six-game unbeaten run, did not lie down and their determined approach limited further City chances.
Sean Dyche’s side had few sights of goal themselves, but it took two goals in the space of two second-half minutes from Nicolas Otamendi and Leroy Sane for the home side to make sure of the points.
Otamendi rose highest from Sane’s corner to power a header into the corner, despite the presence of Steven Defour on the post.
Sane galloped on to a precise through ball from Kevin de Bruyne to score City’s third goal, their 20th in their past four home Premier League games.
City, two points clear before the game, extended their lead over second-placed Manchester United, who lost 2-1 at Huddersfield Town.
