I quote this helpful answer from Quora. In the interest of readability, I forgo quoting with
> and edit the formatting lightly.
[TL;DR; Headnote] Eleven and twelve probably came from words meaning 'one-left' and 'two-left' -- left over, that is, after counting to ten. One explanation for the departure from the 13-19 pattern is that the Old Germanic ancestor languages of English could essentially only count up to ten, with hacks for numbers just over ten. The more computational system of the -teens later came to augment -- but did not replace -- the two ancient numerals.
There are two sub-questions:
- What are the etymologies of the words eleven and twelve?
- Why don't these words fit the pattern of thirteen through nineteen?
[Answer to 1] A remarkably thorough answer to (1) appears at Why do eleven and twelve get unique words and not end in "-teen"? . (The part of the answer that begins "Edit: On further research..." is better than the main part of the answer.)
In brief, the answer is that eleven and twelve probably do fit a pattern -- a different one from the -teen words. Eleven is derived from Old Germanic *ainlif-, which is a combination of *ain-, meaning one, and -lif-, of uncertain origin . A theory the Oxford English Dictionary ventures about the origin of -lif- is that it is cognate with Germanic words meaning "left" (in the sense of "remaining"), so that eleven is "one left" (that is, one remaining after counting ten), and twelve is, analogously, "two left". The theory that -lif- has something to do with ten is considered discredited by OED but not by all etymological dictionaries. There is not much etymological support, it seems, for the theory of a base-twelve origin of these words.
[Answer to 2] [...] 2 [, rephrased, becomes:] Why are 'eleven' and 'twelve' constructed according to a different pattern? 2 is a tantalizing question. I'll quote a speculation of Karl Menninger from the book Number Words and Number Symbols: A Cultural History of Numbers . Menninger also subscribes to the "one left" and "two left" theory of eleven and twelve. He continues:
This is striking evidence that the Germanic number sequence at one time ran only as far as ten. Anything above that was "more". One and two more than ten were still counted, but anything beyond them was perhaps, as so often happens among primitive people, merely considered "many." Then along with the later clear conception of numbers, the subsequent computational number-word formations arose: 3'10 thir-teen, 4'10 four-teen, 5'10 fif-teen, and so on. (p. 84) 
Later, Menninger notes that "[n]umber words are among the words of a language that most strongly resist change" (p. 100). Thus it makes sense that the new, more logical/computational -teen words would augment but not replace the old "one left" and "two left" words. The durability of number words also supports the hypothesis that an extension of the "N left" system was not widespread in the ancestors of English, because if it had been, it probably wouldn't have been widely superseded by the -teen words.
Still, there is something a little odd about the theory that speakers of Old Germanic did not follow the pattern and start saying "three left", etc. -- that they used a hack to count slightly over ten, but stopped far short of using that idea to its full power. Such non-pattern-following would seem to correspond to a pretty early stage of intellectual development in the history of the Germanic tribes -- presumably before they came into regular contact with Romans, who knew how to count much higher. Incidentally, Lithuanian is the one Indo-European language known to actually have continued the pattern ("three left", "four left", all the way up to "nine left"); it has kept these words until today. Menninger (p. 84) says there is "no doubt" these number words were brought to the Lithuanians by migrating Germanic peoples; it is not clear whether the Germanic peoples brought only the start of the pattern (eleven and twelve) or the whole thing. If the latter is the case, then the observation at the end of the previous paragraph suggests that this was a fairly isolated Germanic innovation, one that did not take hold in many places -- otherwise, the -teen words would have had a hard time taking over.
Overall, our knowledge of the answer to question (2) is obviously much sketchier than of the answer to (1), but it is fun to speculate.
 Paraphrasing the Oxford English Dictionary, under the entry eleven adj. and n.
 For another argument that the early Germanic tribes counted barely past ten, see pp. 85-86 of The Numeral Words by Melius De Villiers, http://books.google.com/books?id=KXFV-9w2VWUC&lpg=PA3&ots=My0Gl9DuI3&dq=eleven%20twelve%20germanic%20numerals&lr&pg=PA85#v=onepage&q=twelve&f=false . He also gives examples to suggest that in several other languages, just as in the West Germanic ones, the words for eleven and twelve are older and more primitive than those for 13-19.