Aguero's landmark moment came from the penalty spot on the half-hour mark as he converted his 177th goal in City colours, before a quickfire double from Nicolas Otamendi and Leroy Sane took the game beyond the visitors.
City have now equalled their club record of 11 consecutive victories across all competitions, with the latest in that run taking them five points clear at the top of the Premier League table after Manchester United fell to a shock defeat at Huddersfield Town.
City began with the type of swagger expected from a team who had scored 29 goals in their opening eight games - the best top-flight scoring record since 1894 - but they found it difficult to break through Burnley and their only sight of goal during the opening exchanges saw Nick Pope save a tame Kevin De Bruyne effort.
Wood picked up an injury in the collision which forced him off, though, and City soon had their first chance of the contest when Sane latched on to David Silva's through-ball before reaching the byline and pulling the ball back for Bernardo Silva, who fired his effort straight at Pope when he should have done better.
Pope was called into action again shortly before the half-hour mark when he was quick off his line to thwart De Bruyne, although seconds later he was late to the ball when challenging Bernardo and referee Roger East ruled that there was sufficient contact to award a penalty.
Aguero stepped up to take the spot kick, and made no mistake by sending Pope the wrong way to draw level with Brook's 177-goal mark.
The Argentine soon set about chasing the outright record, and he flashed one effort wide of the far post before being denied twice in quick succession by the feet of Pope.
Burnley reached half time with just a one-goal deficit, though, and they continued to make things difficult for the league leaders during the second half as City struggled to find a way through a stubborn Clarets defence.
The visitors almost levelled things up when James Tarkowski failed to make the most of finding himself unmarked inside the area, and all City could muster in response prior to the final 20 minutes was a sliced Silva strike which flew well wide.
Silva did draw a save from Pope with a little over 15 minutes remaining, though, and that sparked City's quickfire double as they took the game away from Burnley in the blink of an eye.
The resulting corner from Silva's effort provided the first as Otamendi climbed highest in the area to plant his header home, with Steven Defour unable to prevent the ball from sneaking over the line.
Having netted from a penalty and corner so far, City then scored in a fashion more akin to their usual style when De Bruyne split the defence with a fine through-ball for Sane, who cut inside before curling his effort past Pope.
Suddenly City looked like improving their incredible goalscoring form as they chased a fourth, and Gabriel Jesus - on for the record-equalling Aguero - should have provided it when he was denied by Pope from point-blank range after Sane and Silva had failed to get on the end of Walker's low cross.
Sane squandered another late chance to add to the home side's lead in stoppage time as Burnley avoided any further damage, but by that stage City had already wrapped up their 11th consecutive victory to enhance their title credentials even further.
Burnley, meanwhile, drop two places to ninth in the standings following their first league defeat since August.