Would you describe the thrashing of Hungary as by the "skin of their teeth"?
The Mongols don't need to counter heavy cavalry with heavy cavalry of their own. They can just take advantage of the fact knights have about the same sense of discipline as berserkers.
Plus 13th century knights weren`t that heavy yet (mail only, great helms just started to appear), and being formed by small social elite they weren`t so numerous, compared to Mongol horde which mobilised whole tribes for an invasion.