Gorgeous George is the name I've heard/read here and there get tossed into the mix but for years I always thought of Hayes/The Free Birds as the answer. If they technically weren't THE first, they most certainly were the ones that started the trend of it be the new standard and creative way of presenting wrestlers.
Of course with the WWF Rock n' Wrestling period in the mid-80s, entrance music was tied to the business forever from that point.
Btw, first wrestling album ever from 1985:
