The history of the idea underlying the short/long/synthetic division turned out to be far more complicated than I expected, somewhat reminiscent of the history of $0$, with no single inventor. According to the Angelfire timeline the modern long division symbol of English-speaking countries is first used in the 1888 teacher's edition of The Elements of Algebra by G. A. Wentworth. However, as Wikipedia attests, its use is far from universal even today. The arrangement of calculations grew out of two much earlier preceding methods, the Italian and the galley. Originally, the divisor was written on one side of the dividend, and the quotient on the other. Frank Swetz suggests in Capitalism and Arithmetic that the quotient remained on the right by custom after the galley method gave way to the Italian method in the 17th century. And only with the advent of decimal division, and the greater need to align the decimal places, the quotient was moved to above the dividend.
A detailed account of early paper-and-pencil division algorithms is given on Pat Ballew's blog, not all of them are "equivalent" to long division even speaking broadly, and many of them survived into 20-th century. For example, one method described by the famous Fibonacci in his Liber Abaci of 1202, required prime factoring the dividend first. The closest predecessor of the modern long division is the Italian method, which simply omits writing the partial products, so it is closer to the short division. It was first described by Calandri in a 1491 book, and nicknamed "danda" ("giving") by Cataneo in 1546. Cataneo noted that during the division process, after each subtraction of partial products, another figure from the dividend is ‘given’ to the remainder. Henry Briggs, the first professor of geometry at Gresham college is credited with transforming danda into long division in early 1597.
Danda is more economical in terms of paper space, an important consideration at the times when paper was expensive, but the galley or scratch method was even better on this score, as parts of the calculation could be successively erased and written over. According to Cajori's History of Mathematical Notations:"It will be remembered that the scratch method did not spring into existence in the form taught by the writers of the sixteenth century. On the contrary, it is simply the graphical representation of the method employed by the Hindus, who calculated with a coarse pencil on a small dust-covered tablet. The erasing of a figure by the Hindus is here represented by the scratching of a figure". Apparently, Al-Khwarizmi, the founder of algebra, also used this method.
Some authors credit medieval Arabic mathematician al-Samawal (1130-1180) for inventing long division, see also Victor Katz's book (7.2.3). Al-Samawal was first to use tables of coefficients to write and perform calculations with polynomials, he even allowed negative powers. He described a division method underlying the polynomial long division, although his record keeping is more akin to what is now called synthetic division. He also noted the analogy between his way of writing polynomials and the decimal positional notation, and transferred his algorithms to decimal numbers by simply replacing the variable with $10$, so he may well be the first to provide a mathematical justification for a positional division algorithm.
There are even older twists to the story. Many advantages of positional notation are delivered by the abacus, a manual counting device representing numbers by successive rows originally filled with beads or pebbles, which has the added advantage of needing no paper at all. Gerbert improved the abacus by placing symbols at the top of each column, and developed a division algorithm for it, first in Europe, around 972-982 AD, while working at a cathedral school at Rheims. Gerbert is better known for being the only mathematician to be eventually elected the pope, Sylvester II. Division on the abacus became widely known in Europe long before the Arab numerals were introduced. See how an abacus algorithm compares to long division.
The abacus itself goes much further back. Sumerian abacus appeared as far back as 2700–2300 BC, its successive rows reflecting the successive orders of magnitude of the sexagesimal number system. Babylonians likely derived their addition and subtraction algorithms from abacus, and developed one for multiplication. However, according to Victor Katz's book (1.2.2), they had no division algorithm, and used tables of reciprocals and multiplication instead. So long division doesn't go that far back. It is unclear when a division algorithm for the abacus was first invented, a likely place is China after 190 AD. Chinese abacus is called suanpan, and originally used the hexadecimal system. Roman abacus is sufficiently similar to suanpan to suggest transmission, albeit unconfirmed, it is the Roman abacus that Gerbert improved upon.