The result left the Bosnians in a good position to clinch at least a play-off berth in Group D, having collected seven points from four matches after Dzeko headed home an 83rd-minute winner in front of a capacity 15,000 crowd on a bumpy pitch.
Their quest, however, could be derailed by a squabble in the Bosnian Football Federation (NFSBiH), which has been given an April 1 deadline by UEFA to replace its three-man rotating presidential system with one in line with FIFA and UEFA standards.
A Congress the NFSBiH has scheduled for Tuesday is the body's last chance to adhere to UEFA's instructions and avoid suspension from international competition for the Balkan country's national team and clubs.
The Bosnians seemed unbothered by the prospect that they might be playing their last match in the Euro 2012 qualifiers and created several good chances in the first half, but it was the visitors who scored against the run of play.
Having survived a fierce onslaught at their end, Romania took the lead with a goal of superb individual quality when striker Ciprian Marica chested a deflected cross and gave Bosnia keeper Kenan Hasagic no chance with an acrobatic volley.
Adrian Mutu missed a sitter to double Romania's lead shortly after the break and they were punished several minutes later when Vedad Ibisevic glanced a teasing Zvjezdan Misimovic cross into the far corner.
Manchester City striker Dzeko, who is yet to score a Premier League goal for his club, then sent the passionate home crowd into raptures as he ghosted into the goal-kick area box and headed a rebound into the empty net.
Bosnia are due to play their next match away to Romania on 3 June.
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