Sead Kolasinac's unfortunate own goal in just the third minute sent the South Americans on their way before Lionel Messi scored with a fine second-half effort at the Estadio do Maracana.
Substitute Vedad Ibisevic gave Bosnia hope with an 85th-minute goal, but they were unable to find an equaliser.
Alejandro Sabella would have been worried about the opening hour, but his team finished decently after Messi's second World Cup goal and first since 2006.
Starting with a 5-3-2 was perhaps the biggest surprise to Sabella's Argentina line-up.
But Safet Susic's men made the worst possible start to be behind after two minutes and eight seconds.
Messi's set-piece from the left was headed by Marcos Rojo, hit a stranded Kolasinac and crept past goalkeeper Asmir Begovic for the fastest own goal in World Cup history.
Izet Hajrovic almost had Bosnia level in the 13th minute, but goalkeeper Sergio Romero scrambled away after the midfielder's heavy first touch.
Despite the awful start, Susic's side looked comfortable in defence and were just lacking some polish in the attacking third.
Maxi Rodriguez blazed well over from 30 yards just after the half-hour mark for Argentina, while Javier Mascherano's effort from range forced Begovic into a save moments later.
Senad Lulic almost had Bosnia's equaliser from a Miralem Pjanic corner, only for Romero to push the header around his post in the last chance of the half.
Unhappy with his side's first half, Sabella made two changes at the break as Higuain and Fernando Gago replaced Rodriguez and Hugo Campagnaro.
Sergio Aguero's strike from an angle lacked the power and direction to seriously trouble Begovic after he was teed up by Messi.
Messi had squandered a good free-kick chance before he doubled Argentina's lead in the 65th minute.
Ibisevic pulled a goal back for Susic's side as he poked through Romero's legs after a fine through ball by Lulic, but Argentina held on.
