The metaphor of a knot is one of binding, as two people are bound together in marriage.
Shakespeare uses the metaphor, but not the exact phrase:
Send for the county. Go tell him of this.
I'll have this knot knit up tomorrow morning.
But it's earlier still, in The Legend of Saint Katherine c 1225:
Swa wit beoð ifestnet & iteiet in an, & swa þe cnotte is icnut bituhhen unc tweien.
Or as a rough stab at a translation into modern English:
As we are fastened & tied together, so the knot is knitted between us two.*
It may relate to knot-tying as an actual part of wedding ceremonies (and sometimes betrothal ceremonies), as is found in rituals from throughout the world from ancient times until the present day.
Or it may just be a metaphor applied in the phrase alone.
Being so old, it's probably impossible to tell which.
Either way, with knot being in English for so long as a symbol of marriage, and tying being how one forms them, "tie the knot" was pretty inevitable from that starting point.
*Some notes on translation, since they might be of interest in their own way:
I'm tempted to have it "...us twain" rather than "...us two" but while twain is Modern English, it's not common in Contemporary Modern English. It would be a closer translation.
A little is lost in translating wit to we and unc to us as Modern English doesn't have pronouns that specifically refer to two people. "We two" and "us two" would be more accurate but more clumsy.
Knitted would be a rare choice for a knot today—not unheard of, but rarer than once was the case (e.g. see the Shakespeare quote as an example in Early Modern English). Tied or fastened would perhaps be a better translation for that reason, but it would introduce a repetition that scanned a bit silly to my eye.
I'm not happy about the translation of "in an" as "together". "In one" or "as one" would be a closer translation, and more poetic in allusion, but strange with the rest of the sentence in Modern English.
Similarly, swa is the root of the Modern English so but also used as we would use that, then (consider how Irish English still uses so where other dialects would only use then) and as. This mean that the form "as ... as ..." (e.g. "as white as snow") was more flexible in Old and Middle English than today. Translating sadly loses the poetry of the repetition.
Best Answer
The OED has it from 1941 and originally Australian.
They say:
Duff n.1 b* is a flour pudding that's been boiled in a bag, or a dumpling. The a sense is simply dough. Of this they say:
The duff etymology is of northern England, dating to 1840. How did this get to Australia? Well, duff does appear to have been in use in Australia too, such as in the cake known as plum duff which can be found in Australian newspapers as far back as 1849.
An 1893 article relates the naming of this "great holiday dish of sailors" by a "boatswain's mate, a brawny son of the Emerald Isle" whose task it was to cook a Christmas pudding:
Well, this is probably folk etymology, but this sailors' cake is still known as plum duff in the antipodes today, and it's one of many examples of dough -> duff in 19th century Australia.