The Oxford English Dictionary links the expression to talk in the sense of "[t]o say something to the purpose", which has Dickens (1841) as the earliest example:
That's the kind of game... Now you talk, indeed!
The OED (page last updated 2002) attests "money talks" as far back as 1880, but I found earlier examples:
☞ C. P. Hiller, Esq., the agent of the Ætna Insurance Company, has paid Mr. A. I. McCrea five thousand dollars, being the amount of his insurance, in that company, upon the Taycheedeh Mills, destroyed by fire last spring. Money talks. —Sheboygan Journal
Milwaukee Daily Sentinel (Milwaukee, Wisconsin), Friday, June 30, 1854.
Scores of Mr. Moulders [a character from Orley Farm] are to be found in every city about election time, especially among the Democracy. They have nothing to say about the merits of the questions at issue, but they will "back their opinions" to any extent that their party carries particular States. They are boastful that their "money talks" for them. The money of some of them talked so loudly about the October elections that they will be impelled to do their own swearing over their folly.
The Daily Cleveland Herald (Cleveland, Ohio), Monday, October 19, 1868
News By Telegraph
[subheading] The last hours of the canvass—the prospect more cheering than ever—business at a standstill—"money talks"—even betting on ten thousand majority for Hartranft[...]
Boston Daily Advertiser (Boston, Massachusetts), Tuesday, October 08, 1872
I also found an early related example for "money speaks":
M. Codben (M. P.) related a contribution of £100, and said of the donor:
"I tried to have a little talk with him on the corn-laws as affecting his own interests; but I could get nothing from the old gentleman but the simple remark: 'It is a question of money; it is a question of money; and the money speaks for itself.'—(Applause)"
Daily National Intelligencer (Washington, District Of Columbia), Friday, December 09, 1842
It's also worth mentioning that I found a 1884 article entitled "Moujik Wit and Wisdom" which says that there is a Russian proverb that translates to "when money speaks the truth keeps silent". A very, very similar proverb is "Where gold speaks every tongue is silent", which I found in this 1659 book (it is probably the same proverb as the Russian one, although the book does not explain which language any of the proverbs are from). Another book of proverbs from 1659 (maybe the two are related?) lists the Italian proverb "Where gold speaks, every toung[e] is dumb". This could very well be a coincidence, since I haven't any of these proverbs used at all in English and the meaning seems to be quite different.
Digging into older texts, I can find more examples of "money speaks", but there's just not a continuity to link it to what I found in the 1800s newspapers. For example this 1533 text:
mony spekis mekil and rwsis thayme of faith
Translation: Money speaks mickle [greatly] and rooses [boasts] them of faith