Social Darwinism might not be the right word. I mean, during the period when Europeans believed Europeans to be superior to other races by virtue of being European, how did historians explain the Mongol victories in Eastern Europe? I suppose the same question could be asked of the Huns, Persians, Arabs, Turks, and other groups that have fought and defeated the Byzantines and other Eastern European powers.
It will be readily apparent in virtually every history of the Mongols more than ten or fifteen years old. While much of the modern trend is to have a mixed-towards-pragmatically-positive view of the Mongols, the historiography for centuries (particularly out of Russia) was to view the Mongols as a temporary aberration in the course of civilization that really fucked things up - basically, the Mongols succeeded by sheer brutality but were so barbaric that their empire collapsed because they basically were unable to rule it.
The phrase which is ubiquitous in this context is the "Tatar yoke" or "Mongol yoke," which refers to how the Mongols were just so darn awful that they turned Russia into a 'backwards' shithole for centuries afterward. Deconstructing, the term 'backwards' clearly means 'not like Western Europe,' and historians of Russia historically have been loath to tread on the intersection of Orthodox Church and Czar, despite the serious stomping that territory demands in the suppression of Westernization. And it carefully avoids the part where Russia was the most feared nation in Europe before the disaster at Sevastopol. Which is kind of funny, given the USSR, but I digress.
The narrative is to the effect that nasty Mongols murdered a bunch of civilians, the Empire fell apart because they were a 'backwards' people, then they went back to Mongolia for the Chinese to sit on them for a few hundred years. This predates, and postdates social Darwinism, but it fits strongly into the larger narrative. Europeans aren't the best because they're great at killing, they're great at 'advancing' human society. Again, 'advancing' means 'Western Europe' without any real guiding moral principle or rationality.
If you've even skimmed Weatherford's book, or Morgan's Mongols, compare that to this write up on the Mongols. Doesn't sound much like the law-and-order-loving, tolerant, pragmatic Mongols of their books, does it?