Alright, this is going to be borderline ASB, and essentially a China/Persia/Russia-screw, but I think I might have something that resembles such a world.
First, you're going to need to somehow unite the Mongols under one Great Khan. Maybe if that Temujin guy somehow beat Jamukha? He might've been less cautious than his counterpart.
Second, you'll need to have the Song offensives a generation prior fail miserably.
Third, you're going to need to have the Mongols go after the Western Xia before the Jin does and actually win, while at the same time having enough troops to go after Central Asia.
Fourth, you need to have an idiot become the general of the Jin army, complete with mass defections.
Fifth, you'll need to have the Xingsheng emperor assassinated right in the middle of the Mongol invasion, and for his successor to make peace.
Sixth, you'll need a genius to succeed whoever rules the Mongols, and have brilliant generals defect to them.
After that... Maybe if the Song burn all their fleets, and have their entire army commit suicide. Then the Mongols should have a chance.