Raheem Sterling was denied by a fine Dean Henderson stop before City were awarded a penalty as Chris Basham fouled Riyad Mahrez.
But Gabriel Jesus's spot-kick was saved by Henderson, diving to his right.
Pep Guardiola introduced Aguero as a substitute in the 67th minute, and the striker scored the decisive goal six minutes later from Kevin De Bruyne's superb low cross.
Sterling shot wide late on, but one goal was ultimately enough for the champions.
Second-placed City move 13 points behind leaders Liverpool, who have two matches in hand. Sheffield United remain seventh on 33 points.
Sheff Utd: 1 Feb v Crystal Palace (A)
Man City: 2 Feb v Spurs (A)