The African nation now look on course to join Argentina, who they face in their final fixture, in the last 16, while Bosnia, making their first-ever appearance at the global showpiece, will tackle Iran with nothing to play for after defeats in both of their opening games.
Stoke striker Odemwingie scored the only goal of an open contest at the Arena Pantanal when he fired a first-time shot beyond Britannia Stadium team-mate Asmir Begovic after Emmanuel Emenike was given the benefit of the doubt by the officials as he left Emir Spahic in a heap on the floor in the build-up.
Edin Dzeko had two opportunities in added time to bring his side level, but saw a header easily saved by Vincent Enyeama before the Nigeria goalkeeper produced an unorthodox block to push the Manchester City striker's shot onto the post and away to safety.
The recalled Odemwingie saw his excellent free-kick after seven minutes deflected narrowly wide after Haris Medunjanin had been booked for bringing down Ahmed Musa following a blunder by Spahic, but the officials did not spot it and awarded a goal-kick.
John Obi Mikel fired a shot narrowly wide soon after and then Asmir Begovic was called into action to push away a long-range drive from Ogenyi Onazi.
Bosnia responded and Dzeko had the ball in the net in the 21st minute after being played through the centre of the Nigeria defence by Miralem Pjanic, but the assistant referee’s flag was raised for offside, only for replays to show the Etihad forward was clearly not beyond the last man.
That sparked a spell of pressure from Safet Susic’s side, with Izet Hajrovic firing in a fine low shot and Dzeko forcing a good save from Enyeama, before the Nigeria goalkeeper was much less convincing when dealing with Pjanic’s routine effort from range.
The opening goal came in the 29th minute and, again, there were question marks over the role of the officials as Emenike left Spahic on the ground as he out-muscled the Bosnia skipper before bursting to the line and pulling the ball back for Odemwingie to fire beyond Stoke team-mate Begovic.
Susic’s men were incensed that the goal was allowed to stand and could not force an equaliser before the interval, with Dzeko only finding Enyeama with a close-range header before flashing horribly wide in added time following a good pass from Senad Lulic.
The excellent Pjanic played a delightful pass in behind the Nigeria defence to Lulic, but he was unable to fund a team-mate with his pull-back, before Michael Babatunde struck a fierce drive from the right corner of the box which Begovic did well to beat away.
Babatunde worked himself a yard of space outside the penalty area but was again denied by the Stoke man, before Emenike brought the best out of Begovic with a flicked effort from close range which was superbly saved by the onrushing keeper.
Nigeria sent on Shola Ameobi and he was immediately into the fray to play in Emenike, who got away from Spahic with ease yet again and left Toni Sunjic trailing in his wake before firing in a shot which Begovic shovelled around the post.
Substitute Vedad Ibisevic had a glorious opportunity to restore parity in the 74th minute as he was found unmarked on the edge of the six-yard box by Zvjezdan Misimovic corner from the left, but he did not get a clean contact with his header and the ball flew well off target.
Emenike drilled a shot beyond Begovic’s dive but also wide of the post as the game entered the final 10 minutes, and despite Bosnia throwing everything and everyone forward in the closing stages they could not find the goal to prevent an early exit, with Dzeko heading tamely at Enyeama as the game entered added time and then seeing a shot on the turn strike the woodwork.