Pep Guardiola's side dominated the game but struggled to break the deadlock until Sergio Aguero's scuffed shot crossed the line by a matter of millimetres.
Gabriel Jesus came close to adding a second but was denied by a brilliant Ben Mee block.
The champions were unlucky not to have taken the lead earlier in the game after Ashley Barnes clearly handled Bernardo Silva's shot.
City now only have to defeat Leicester and Brighton to clinch back-to-back title wins.
Player Ratings: Burnley vs Manchester City
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