Despite fielding five defenders in St Louis, Argentina continued to show frailties at the back but looked impressive in attack thanks to Rodrigo Palacio and, in particular, Aguero, who scored either side of half-time.
During the first period, Bosnia gave Argentina goalkeeper Sergio Romero plenty to worry about, but they needed some good work from their own stopper, Asmir Begovic, to keep them in the game.
Strikers Vedad Ibisevic and Edin Dzeko both failed with one-on-one chances after mistakes from Argentina defenders, while Begovic produced a top save to deny Aguero when he was through on goal.
Five minutes before the break Begovic was in the action again as he stopped Palacio's shot, but Aguero was on hand to convert the rebound.
Argentina controlled the second half and they doubled their lead after the hour mark when Aguero connected with a superb volley inside the box.
Alejandro Sabella's side had plenty of chances to extend their advantage further but in the end had to be content with two.