I think one of the biggest advantage of the Mongols was that they were basically pow surprise. They had new tactics which were not expected by the Europeans. Hit and run was considered cowardly and such so the Europeans refused to use them, and as a result never develope an counter strategy. However, Mongols were the master of hit and run and feint retreats and such, so basically they got pretty surprised.

In my opinion the real achievement of the Mongols was being able to defeat the Jin and Song Dynasty in such short time, under the conditions of southern China which was not conducive to mongolian warfare. (Huge cities, waters, hills, etc)