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The brief answer would be that the woman triumphs by killing the flea that the speaker has been begging her not to kill. We can infer from the poem that she has killed it with one of her nails. We know this because she is said to have "Purpled thy nail in blood of innocence..."
I think you can also think of this as the woman triumphing by rejecting the man and his desires. He has been hoping that she will not kill the flea and you can see that as a symbol for him hoping she will make love with him. So you might say that she triumphs by rejecting his physical advances (or by squashing his argument about why she should make love with him) as well.
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