Sead Kolasinac scored an own goal in the second minute to get Argentina off to the perfect start, while Lionel Messi added a second after the break.
A late goal from Vedad Ibisevic gave Safet Susic's side hope in the dying minutes, but Sabella's men were able to hold out to secure the victory.
"They were two different halves," Sabella told reporters. "In the first one I think we controlled the Bosnians well, but beyond that, we didn't create the kind of depth that we managed in the second half. Obviously we need to improve.
"It is true that we failed in parts, in part because of my own mistakes."
Argentina face Iran in their next match on Saturday.