Peter Odemwingie's 29th-minute goal lifted the African champions to four points from two games, two less than Argentina, who they meet in their final Group F match on Wednesday with top spot at stake.
Iran are third in the standings with one point and could reach the last 16 if they inflict Bosnia's third defeat in a row in their final group match and Nigeria lose to Argentina.
Nigeria took the lead when Odemwingie fed Emmanuel Emenike on the right wing and raced into the box to receive the striker's low return before sweeping the ball under goalkeeper Asmir Begovic.
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The Bosnian bench angrily protested that Emenike had fouled captain Emir Spahic as he bundled past him into the box before making his cross.
The tournament newcomers had earlier been denied a goal after a poor off-side decision against striker Edin Dzeko after he ran on to a through pass and beat goalkeeper Vincent Enyeama in the 21st minute.
Dzeko had another shot saved by Enyeama but he was well off target with chances later in the half as he fought a lonely battle with the Nigeria defence.
The more experienced Nigerians, playing in their fifth World Cup, created several chances in the first half but were let down by the final pass, poor control or wayward shooting and a couple of smart diving saves by Begovic.
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The Bosnia keeper, a team mate of Odemwingie's at Premier League club Stoke City, made a string of saves, denying Emenike and midfielder Ogenyi Onazi to keep his side in the match in a second half of end-to-end excitement.
Dzeko nearly stole an equaliser for Bosnia, his shot in a goalmouth melee hitting Enyeama's legs and coming back into play off the post deep into stoppage time.
Nigeria coach Stephen Keshi was rewarded for his adventurous tactics while Susic packed his midfield as he had done in the 2-1 defeat by Argentina and only sent on a second striker in Vedad Ibisevic, whose best headed effort went over the bar, for the last half hour.
Just before the Nigerian goal, Enyeama had parried a shot from midfielder Miralem Pjanic and the ball rebounded into the middle of the box but no Bosnian player was following up to take advantage.
Image: Peter Odemwingie scores for Nigeria past Asmir Begovic of Bosnia and Herzegovina during the World Cup Group F match.
Photograph: Clive Brunskill/Getty Images