Well, aside from the very start they could tap the considerable Chinese expertise in siege warfare - given that this tended to be a field nomads were generally somewhat lacking in, that must have been a pretty useful boost.
By what I've read that's also what Khwarimzam fell to - the Khwarimzamshah apparently wasn't aware of the now quite considerable Mongol siege prowess, so by the point he realized sticking around in fortified places was not as good a plan as he'd thought the Mongols had already dealt with much of his forces piecemeal. Had he taken to the field against them from the start he might've come out the winner - AFAIK the Khwarimzam army was very good at fighting, quite large and well equipped and being at least seminomadic themselves no doubt familiar with the tactical chemes the Mongols also liked to use.
'Course, the Mongols had some seriously high-caliber warlords around but then that'd have been the old question of who knows his stuff better.
In any case he did not, so the point is moot.
In principle defeating the Mongol army in war would not require more than a military system familiar with the nomad way of warfare and capable of mustering and supplying enough troops to take them on, or alternatively a sophisticated enough system of fortifications to simply bog the buggers down in endless sieges. So you'd need basic familiarity with the military principles involved, sophisticated enough military structure to mobilize and maintain sufficient numbers of men, and ultimately enough ability in the leaderhip level to out-general the opposition which when you think about it tends to be something of a basic prequisite anyway.
Alas, most of the folks the Mongols ran into lacked at least one of the above.
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