City's Manchester derby win on Wednesday saw them seize the advantage in the title race but as Liverpool thrashed Huddersfield on Friday night, the defending champions went into Sunday's clash at Turf Moor in second spot.
Burnley offered little going forward but did manage to frustrate City until just after the hour mark when Aguero finally forced a breakthrough for Pep Guardiola's men.
The striker's close-range effort was confirmed to have gone in by goal-line technology, despite a last-ditch attempted clearance by Matt Lowton.
There was one solitary nervy moment for City when Chris Wood broke clear in the first half but a poor touch to let the Clarets striker down.
City pushed for a second after Aguero's opener and Gabriel Jesus came close but was denied by a fantastic block from Ben Mee on the line.
The game against the Clarets was earmarked as a potential banana skin by some pundits in the build-up to the clash and Guardiola will have been relieved to have seen City emerge with all three points.
As the game entered the final stages the visitors were uncharacteristically happy to keep possession and sit back in a sign they are feeling the pressure in this intense title fight with Liverpool, although Burnley were unable to create any real chances to equalise as the clock ran down.
City face Leicester at home next on Monday May 6 and may have been knocked off top spot by the time the clash against the Foxes kicks off as Liverpool travel to Newcastle on Saturday.
The champions complete the Premier League campaign at Brighton, while the Reds host Wolves in the final round of fixtures.