The Nigerian was back in action after an ankle injury and made an immediate impact with a neat left-footed finish inside the opening three minutes at Carrow Road, making the most of a defensive blunder by Canaries defender Ritchie De Laet.
Odemwingie admitted that it was important for the Baggies to take maximum points back to The Hawthorns after three losses on the trot.
"It was good to get three points, especially away," he told Sky Sports.
"(After) three defeats, it's not easy going into an international break with our heads down but we improved the atmosphere on the training ground and I'm very pleased."
Odemwingie said Norwich proved tough opposition as they played with the mentality that got them into the top flight and he also made sure he apologised to his teammates for his 75th penalty miss which would have wrapped up proceedings for his side.
"Yes of course. They are team who have a lot of confidence. They came up to the Premiership and that happiness and desperation is what they still have which makes them perform so well," he said when asked if Norwich were difficult to play against.
"I made it difficult for my teammates. I was saying sorry sorry at the end of the game (after Declan Rudd saved the spot-kick), I could have made it easier but that's football, that's why it is exciting. I was saying 'now, let's defend as a team, one of your teammates has made a mistake' and that's why we are a team." "For me personally of course, it's my first start of the season. It put me straight away where I need to be mentally and I'm sure my teammates were happy for me. All 10 of them came to congratulate me and it gave us the confidence.
"We forgot about what happened last three games and right away we were in the game."