You're right. But finally, at the beginning of XIII century the empire divided into Golden Orde, Chagatai Khanate, Yuan dynasty and other states. It was caused by the civil war, right?
What if there was no civil war 1260-64?
Actually, all of those Khanates were sub-organizations of the Mongol Empire. Autonomous "states" that were given clear rights to conquer and expand where they wished. Most of the Mongol Conquests not directly carried out by Genghis Khan were carried out by these Khanates. The division was there from its inception.
Keeping it together requires strong leadership (Mongke was the last effective Khagan of the entire Mongol Empire, with Kublai and a few of his descendents exerting some control over the Il Khanate alone), cultural unity (this was starting to break apart because of the various people that were conquered, only maybe a few ten thousand of the people in these khanates were Mongolian, a few hundred thousand Turkic, the rest natives), and a lack of religious conflict (the Mongols were excellent when they were largely above religious problems, or slightly leaning to Buddhism or Christianity, once they made the plunge to strict Islam-Buddhist divides the khanates were irreconcilable until conversion).
So, you need a Khan's Khan, a Mongol Khan's Khan ruling over Mongol Khans, and a Mongol's Khan's Khan who could enforce tolerance for all religions amongst his religiously diverse or secular Mongol Khans.