Of all seventy-two spirits, this one has by a wide margin the best documented pedigree, and it is worth being precise about what that pedigree does and does not establish.
Asmodeus appears by name in the Book of Tobit, a work of the Second Temple period preserved in Greek and, since the discoveries at Qumran, known also in Aramaic and Hebrew fragments. In it he is a demon who kills seven successive husbands of a woman named Sarah on their wedding nights, and who is driven off by the smoke of a fish's heart and liver and then bound in Egypt. Tobit is canonical for Catholic and Orthodox Christians and deuterocanonical or apocryphal for Jews and Protestants, so the book's standing varies but its antiquity does not.
The name is generally connected to the Avestan aeshma daeva, a demon of wrath, which would make it one of the clearer traces of Iranian influence on Jewish demonology in the Persian period. That derivation is widely held and is argued rather than proven; it rests on the phonetic match and on the plausibility of the contact, not on a text that records the borrowing.
Asmodeus also appears prominently in the Testament of Solomon, and in rabbinic material as Ashmedai, notably in a Talmudic narrative in which he displaces Solomon from his throne.
What none of this establishes is continuity with the Goetia. The Goetia's Asmoday has three heads, a serpent's tail and goose's feet, rides a dragon, teaches arithmetic and geometry, and guards treasure. The name is ancient. The figure in this catalogue is not, and the intervening chain of transmission runs through the same late medieval and early modern channels as the rest of the list.