No, I don't believe "She won the price." is an expression people would normally use. Perhaps in a transaction where they can't decide the price, so decide by flipping a coin or doing 가이 바이 보.
"Let's flip a coin. If I win, you pay me $10 for this LA Dodger cap that you want. If you win, you pay me $9.
OK, the coin came up heads. You won the price.
But that would definitely not be a normal usage.