City won on the road for the sixth time in seven away Premier League games this season despite Dean Marney opening the scoring with an excellent 14th-minute volley.
Pep Guardiola's side lacked the intensity and fluency that marks their best work and were indebted to Aguero's poaching instincts.
The Argentina international levelled when Burnley failed to deal with a corner before the break and was the beneficiary of another penalty-box scramble with an hour played, turning home Fernandinho's cutback.
But City's captain saw plenty of positives from the way the team battled to a win, telling BT Sport: "We analysed Burnley before the game and saw they play for second balls and they scored their goal like that.
"Fortunately we turned it around to score two goals and win.
"I think that is the spirit we need. If you want to be a champion you have to play games like this. Fortunately we calmed down after the first half."
Fernandinho was once again named alongside his old midfield partner Yaya Toure, who retained his place in City's Premier League line-up having returned to hit a brace in last weekend's 2-1 triumph at Crystal Palace.
With a stand-off between his agent Dimitri Seluk and Guardiola now apparently consigned to history, Toure is enjoying an unlikely reprise to his City career that has gone down well at home.
"It was a hard physical game and I think we responded very well. We are happy for the three points again," the Ivorian said.
"I am very delighted to play. I always want to play and always want to help my team-mates. Like my wife said, I am looking much younger."