The Manchester City striker scored in both halves but should have been sent off before his second goal for lashing out at Bosnia defender Sead Kolasinac just before the hour-mark.
But the referee only showed Aguero a yellow card and the 25-year-old forward wrapped up Argentina's win with a thumping strike five minutes later, as Alejandro Sabella's team notched just their second victory since they secured World Cup qualification.
Argentina qualified for Brazil 2014 in September with a 5-2 win away to Paraguay but after beating Peru 3-1 in their next international, Sabella's side lost to Uruguay in their final World Cup qualifier and drew with Ecuador in New Jersey on Friday.
Monday's game at Busch Stadium started in frantic fashion with Rodrigo Palacio and Aguero both failing to convert chances from close range before Bosnia's Vedad Ibisevic missed a one-on-one opportunity, all within the first 10 minutes.
Aguero's Manchester City team-mate Edin Dzeko was next to miss in an open start to the game, as his shot across goal squirmed wide of the far post.
As the first half wore on, Argentina started to gain the ascendency and deservedly hit the front in the 40th minute. Angel Di Maria's header over the top of Bosnia's defence found Palacio and although the Inter striker failed to beat Asmir Begovic, Aguero thumped home the loose ball.
The referee erred in not showing Aguero a red card in the 59th minute as the Argentine striker stamped into the back of Kolasinac's buttock after the Bosnia defender had brought him down in the box.
Aguero took advantage of the referee's leniency in the 64th minute, running onto Maxi Rodriguez's clever chipped pass and crunching his half-volley past Begovic.
Bosnia kept attacking for the remainder of the match with Dzeko sending a 30-metre shot wide of the goal, while Argentina had two more big chances to score with Aguero scooping the ball over the bar following Begovic's save from Palacio and Erik Lamela missing from close range.
But despite the chances, both sides failed to score and Argentina claimed victory in their last match of the year.