Goals from Anthony Knockaert and Yves Bissouma put Albion 2-0 up at half-time but Bournemouth got a goal back through Marc Pugh after 55 minutes.
Andone then got the third nine minutes later to seal the win and the Romanian forward said afterwards: "For me, it was very important. For strikers if you score, you have more confidence.
"It doesn't matter if it's the Premier League, FA Cup or if it's in training, I try to score always. Today is a perfect day, we won, I scored and I'm very happy."
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On the match, Andone said: "We deserved the victory.
"We were very focused and we were better than them. This is the way - to keep fighting and working every match to have this feeling of winning.
"I'm very proud of my team-mates and this is the way we need to follow.
"Momentum is very important. This is a different competition but, for us, it is just one game more."
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