Note that when the Mongols conquered China, they used Chinese clerks to do their record keeping.
Having a Mongol dynasty make CHINESE the world's lingua franca would be 'easy'. Making Mongolian that, much, much, much harder.
How early are Mongols even an identifiable people, let alone a world threat?
A quick search suggests that they don't really arise until about the 6th century, by which time Chinese culture is well established, Europe is well on the way to Christianizing, etc.
When barbarian pagan conquerors take over civilized countries, they usually assimilate to the local culture. Cf. Mongols in China, German tribes in the ex-Roman Empire, Bulgars and Avars in the Balkans, etc.
To prevent that, you probably need a sufficiently strong religion (e.g. Turks taking over Anatolia, Arabs most of the rest of the Middle East).
So....
Convert the Mongols to Buddhism or Christianity, and have them go Crusading (or local equivalent), and you MIGHT be able to get such a thing happening. If you manage it before Islam arises (or spreads widely), you might get a Mongolian based Persian Empire or even (possibly) Persian and Chinese empires.
Darn close to ASB, though.