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Both sides started the match in St Louis with plenty of attacking intent, forcing early strikes on goal as Manchester City teammates Edin Dzeko and Aguero were thwarted at either end early on.
Aguero was then kept at bay by a strong stop from Asmir Begovic but that save only gave Bosnia a momentary reprieve, with the City striker breaking the deadlock soon after.
Argentina edged ahead after Begovic had blocked Rodrigo Palacio's path to goal, but could only parry out to Aguero as he smashed home from close range five minutes before the break.
Early in the second half Aguero was again involved and was slightly fortunate not to be red-carded after kicking out at Emir Spahic with studs showing.
Bosnia were left to rue a lenient booking as Aguero doubled his tally in some style soon after, allowing a lofted pass from Maxi Rodriguez to sit up invitingly before hammering home on the half-volley.
Substitute Erik Lamela spurned two glorious late chances to extend Argentina's advantage before Aguero fired over with the hat-trick in sight, with Bosnia unable to force a late response.