The Stoke City forward scored the only goal of the game on 29 minutes, but Odemwingie admitted that the African team were made to work for the three points.
He told BBC Sport: "This victory was for our fans and our country. We had to work hard for this against a quality side. It was a difficult three points but we worked hard to bounce back after what our people thought was a poor performance in the first game.
"From now on we are looking upwards. Hopefully this win is enough to get us into the second round and after that to be inspired - like we were in the African Cup of Nations."
Nigeria face Argentina in their final group game this Wednesday.