This article and this article are worth reading for you. "Much of Central Asia" is not correct - less than ten percent, according to the studies we have. There are some probably some caveats about ethnic diversity for those studies, like I would have preferred a study of only nomadic peoples, but there it stands.

If I didn't have links, this wouldn't be a top-level post, because I basically support /u/Valkine.

Chingis Khan might have had a conspicuously large number of bastards, but it's not like he spent years of his life sowing his seed, so to speak. He was a very busy man. Importantly, his descendants ruled much of the Middle East until the 14th century, the Crimea until almost the 19th century, Russia for two generations, parts of China until 1388, most of the steppe into the 1300s, and parts of Mongolia until 1922, should what I've read about Naavanneren be correct.

Chingis had four sons of note; Ogedei, Tolui, Jochi, and Chagatai. They had, respectively, at least, 7 sons, 4 sons, 3 sons, and 3 sons. Virtually all of those men were powerful military and political leaders. Kublai, one of these grandsons, had more than ten sons (IIRC, I'm trying to find a concise source, but I'm not so well-versed in the Yuan, so I apologize).

And then to add to this mess you have 130 years of Timur's empire. Timur was not a descendant of Chingis himself, but there was little doubt that Chingissids were amongst the aristocracy there. Additionally, the three-hundred-year Mughal Empire's founder claimed to be a Chagataid.

With the exception of propensity to alcoholism, Chingis' descendants seem to be quite genetically healthy and physically fit. The Chingissids were powerful, populous, and widespread. There was no need for Chingis himself to sire an incredible number of children, because his genes already had incredible reach.

Genetic history is a super interesting field but one that I entirely lack the expertise to ever properly engage in. However, my university does do some work in the field and I have had the privilege of seeing Prof. Dan Bradley talk about this kind of research. Prof. Bradley primarily does research on cattle genetics but as a side project he does research very similar to this Ghenghis Khan project but focused on Ireland instead of Asia. If you're a fan of puns you'll be happy to know that he has dubbed the target of his research Ghenghis Dan.

I'm not an expert on this particular research but I have some familiarity with a very similar project, from the aforementioned Prof. Bradley. Prof. Bradley's theory was that great families allow for greater genetic survival, essentially boiled down to political power could be related to genetic survival. In this theory Ghenghis Khan wouldn't have had to have dozens of children to spread his genes so far, what is more important is that his children survived. Even more important after that is that his children's children survived, and produced more children etc... In a sense, if we take Prof. Bradley's view which I should stress is not necessarily the view held by other researchers, what would be a more significant factor to the spread of Ghenghis genetic line was the enduring Mongol empire rather than how many children he specifically had. So long as every member of his family who survived to reproductive age was able to find a mate and have more children who in turn found mates to have children with his genetic line would have spread very successfully. The fact that his line was hugely politically powerful made it easier for his family to reproduce and the fact that his family was spread so far across Asia certainly helped spread the genes around a bit.

If you're curious about more related topics to this a lot of Prof. Bradley's work can be read from links found on this page. His research is primarily focused on Y-Chromosomes in Irish families, though, and distinctly lacking in Ghenghis Khan.

He wouldn't have needed to be particularly fecund to be an ancestor of large swathes of the central Asian population, he just needed to produce offspring who themselves produced offspring and who in turn produced offspring... compare Charlemagne who was ruling in Europe a few hundred years before Genghis appeared on the scene - he is credited with being a famous ancestor of virtually all indigenous Europeans alive today - not because he had a zillion concubines - but because mathematics and common sense dictates that it should necessarily be so. For a basic guide the most recent common ancestor of Europeans is believed to have lived just 600 years ago.

You have two parents, probably four grandparents, probably eight great grandparents... and so on.. but the numbers can't double with every generation or else you'd end up with a number of ancestors much, much larger than the population of the entire world at that time. This means that you must share lots of ancestors with other people and that some of your individual ancestors must appear on lots of different places on your family tree. At some point a cousin marries a cousin or a niece marries an uncle and the family tree starts overlapping on itself.

So, the question here is not 'are lots of central Asian people and people around the world descended from Genghis' that's a simple maths question to which the answer is 'yes, if his reproductive success carried through his blood line' the real interesting question is 'has Genghis contributed substantially more of his DNA to today's populations than his contemporaries? Was the gene pool of central Asia narrowed due to his having lived and bred?

http://phenomena.nationalgeographic.com/2013/05/07/charlemagnes-dna-and-our-universal-royalty/