As has been pointed out in the comments, the closest correct answer to this question that I'm aware of is "sophistry".
That is, there is no term to my knowledge that is specifically about winning an argument using elegant/deceptive manipulation, but the act itself is known as sophistry. The people who do this are known as sophists.
Being called a sophist wasn't always meant in a negative way, and indeed the original terms referred to people who were simply well-spoken. The word actually comes from Greek σόφισμα, sophisma, from σοφίζω, sophizo "I am wise", but this changed over time.
Owing largely to the influence of Plato and Aristotle, philosophy came
to be regarded as distinct from sophistry, the latter being regarded
as specious and rhetorical, a practical discipline. Thus, by the time
of the Roman Empire, a sophist was simply a teacher of rhetoric and a
popular public speaker.