City were frustrated for long periods at Turf Moor on Sunday but they found the breakthrough in the 63rd minute, Aguero scoring his 20th Premier League goal of the season.
Aguero's strike evaded the desperate efforts of Burnley defender Matt Lowton to clear off the line, with City's 12th straight league victory moving Pep Guardiola's side a point above Liverpool with two games remaining.
The Argentina attacker has become the second player in Premier League history to score 20 or more times in five consecutive seasons, with his latest goal keeping City on track for the title.
KING KUN pic.twitter.com/O2JaQlczq8— Manchester City (@ManCity) April 28, 2019
City took until the 28th minute to record the game's first shot on target, but Bernardo Silva's attempt to repeat his Old Trafford goal from midweek failed to seriously test Tom Heaton at his near post.
Leroy Sane showed frustration with City's sloppy play in the final third before Aguero thumped a drive from outside the Burnley box narrowly wide of Heaton's left upright.
City, after a tunnel pep talk from Guardiola, made a roaring start to the second half with Heaton beating away Aguero's powerful strike before Ashley Barnes appeared to block Silva's shot with his arm and Aymeric Laporte headed straight at the keeper.
Burnley were on the ropes and City found a way through thanks to Aguero, who has now scored nine goals in his past eight appearances in all competitions against the Clarets.
James Tarkowski appeared to have halted Aguero in the box, but City's record goalscorer wriggled clear and Lowton's second attempt to clear his shot off the line was timed moments too late.
Substitute Gabriel Jesus almost added a nerve-settling second only for former City centre-back Ben Mee to make a brilliant clearance on the line, bettering Lowton's efforts.
Guardiola decided to hold what he had, finishing the match with all four of his senior centre-halves on the field to cautiously see out a vital triumph.
What does it mean? City grind it out
After a 2-0 win against rivals Manchester United in midweek, a trip to Turf Moor might have been considered the toughest remaining assignment for Guardiola's men.
In the first half City looked nervous and failed to work Heaton, but after the interval it was a different story and the opening goal, by the time it arrived, had been coming for some time.
Aguero makes the difference again
It was of course Aguero who scored that memorable goal against Queens Park Rangers that sealed City's first Premier League title, but his winner against Burnley may in time be judged to be almost as important.
His latest landmark, reaching 20 Premier League goals yet again, means he must be judged alongside players such as Thierry Henry and Alan Shearer among the division's greatest marksmen.
20 - Sergio Agüero is only the second player in @premierleague history to score 20+ goals in five consecutive campaigns, after Thierry Henry between 2001-02 and 2005-06. Legends. #BURMCI pic.twitter.com/5o0MSDtcgL— OptaJoe (@OptaJoe) April 28, 2019
Sane struggles to make his mark
The future of City winger Sane continues to be hotly debated but, given a start after his fine clinching goal against United, he did not justify his selection.
Sane did not seem on the same wavelength as his attacking colleagues and he was replaced by Jesus moments after Aguero opened the scoring.