Nigeria’s win, their first at a World Cup finals since 1998, leaves them as favourites to join Argentina in the second round.
Argentina are on six points after beating Iran 1-0 earlier on Saturday to leave the Iranians on a point while the Nigerians are on four points.
Nigeria play Argentina in their final group game on Thursday — the Africans have lost all three times they have previously played the South American side in the group stages, losing 1-0 on the last occasion in 2010.
Odemwingie, who has had his fair share of run ins with national coach Stephen Keshi, said the team had been painfully aware of the hiatus between wins.
“We were aware of the 16 year gap with no win for the country,” said the 32-year-old, who was born in the former Soviet Republic of Uzbekistan where his father played football.
“This is very exciting for us and very important too as the fans were disappointed by our previous performance (a 0-0 draw with Iran).
“This victory can have a huge impact on us as it brings confidence with it and if we get to the last 16 who knows what can happen, maybe a repeat of what happened in last year’s Africa Cup of Nations where we got better and better.”
Bosnia coach Safet Susic, who started with just Dzeko up front despite saying after the opening defeat to Argentina he would play with two strikers, said that his side had been robbed of a goal by New Zealand referee Peter O’Leary.
“I was told there was no offside but it’s not the first or last time a referee has made a bad decision at this World Cup and it won’t be the last,” said the 59-year-old.
“We lost to a good side, Nigeria played better than against Iran.”
Nigeria were, as Susic attested to, a completely different team to the one that had barely got out of a Sunday morning stroll in the draw with Iran.
Emmanuel Emineke always looked a threat and it was he who created the goal, as he burst down the right and engaged in a race for the ball with veteran Bosnian skipper Emir Spahic, who tried vainly to obstruct the Nigerian.
However, the central defender had completely misjudged the situation as Emineke went the opposite side and outpaced him to collect the ball before unselfishly squaring it to Peter Odemwingie, with the Stoke City striker tucking the ball away.
Bosnia had little to offer, though they could claim that they had been robbed of a goal as Edin Dzeko put the ball past Vincent Enyeama, but it was wrongly ruled out for offside.
Dzeko, though, could have stolen a point at the death when he rose to meet a corner but his downward header was straight at Enyeama, who gratefully smothered it.
The Manchester City striker went even closer in time added on as his shot from inside the area beat Enyeama but cannoned back off the post.
The referee whistled for fulltime shortly after that with the Bosnians collapsing to the ground in despair and the Nigerians jumping for joy as they — after the Ghanaians drew with Germany earlier on Saturday — restored pride to the African challenge at the finals.