Was it their tactical skills usually praised for its beyond-age motional aspect and agility? Or rather, was it their highly advanced topographical perception and informational meticulousness giving the advantage?

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This and even more.

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Mongols did have big armies and sometimes even possessed numerical advantage in battles, but this was far from being the decisive factor.

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To understand why Mongol war machine was so effective means to understand why it differed so much from other countries, why exactly were their opponents unable to cope with it. Constant, systematic supremacy of Mongol armies who rarely suffered defeats at this time indicates that they indeed were much more able to beat their adversaries in different situations, including open field battles and sieges, than vice versa.

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In short, war system implemented by Mongols was unique for the time in regards to how troops operate both on strategic and tactical levels.

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On strategic level Mongols were very flexible, they knew exactly where and when to strike thanks to their great mobility. They learned about weak spots of their opponents beforehand, for example, during the conquest of medieval Russia (and later punitive expeditions in the same area) their main combat activities focused on attacking in winter which opened up several key advantages:

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1. Enemy peasants have already stacked up their stockpiles with food and resources which could be easily taken by the invaders.
2. No mud and slush to speak of, frozen rivers provide good alternative for paved roads.
3. Their horses were pretty resistant to cold weather and were actually even more enduring during winter times since they had excessive amount of fat they grew while pasturing in spring and summer months.

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Similar considerations applied to other regions Mongols conquered - they knew how to turn warring factions against each other, knew how to isolate their opponents and strike where they're the weakest. Medieval times are an era of sieges, and Mongols, being taught by Chinese craftsmen, could utilize quite complex siege engines, but their primary objective during invasions was to quickly defeat enemy field armies so that no one would bother them later. They knew that sieges provide luducrious possibilites to die from suddenly appearing enemy reinforcements, illness and defenders' sorties so they preferred to minimize these negative effects. By seeking decisive engagement, Mongols played to their strong sides: they knew they are most likely better in direct confrontation than their opponents and if the enemy army is completely destroyed, then it was relatively easy to take cities and fortresses one by one without further interruption.

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Being excellent horse riders and archers, Mongols tried to utilize their strong sides to the extreme by actively using feigned retreats, harrassing flanks of enemy army and engaging in close combat preferrably when the opponent already breaks and starts fleeing. They could easily outmaneuver their adversaries, surprise them with lightning fast flanking attack, then retreat in order to lure them into the ambush, encircle and later completely annihilate them. This tactic couldn't be brought to life without very strict discipline, which was partly implemented with harsh measures (deserters and cowards were always exectued) and partly with fullscale military exercices called war hunts, where large war parties of Mongols learned to properly attack, regroup, encircle and pursue their targets in forms of wild animals which actually were great substitutes for the enemy soldiers.

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This iron discipline and proper understanding of battle tactics by all soldiers in the army ensued their superiority since their enemies usually lacked these qualities. If you take a closer look at, say, European or Arabic armies of the 13th century, it is quite clear that they didn't possess same levels of competence. Many battles were fought and either won or lost due to sheer luck or random disobedience of soldiers and officers seeking glory, being too arrogant or simply untrained to conduct their actions together. Battle of Arsuf (1191) included notable episode of knight cavalry disobeying direct order not to intervene and attacking Saladin's army because they were extremely impetuous. Similar things happened during the battle of Crecy (1346) when impulsive French knights stroke at English army in completely disorganized fashion even managing to inflict casualties on their own infantry. Same was true for the battle of Lipiza (1216): Russian knights from Rostov and Smolensk turned out to be very impatient and attacked right through their infantry.

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Also note that European nobility and even kings often found their deaths in battle while Mongols valued their commanders and never put them directly on the frontlines - by adequately choosing their command post, Mongol generals such as Subutai could operate with great effeciency. The only example of direct Chingizid royalty dieing in battle I can think of is Kulkan who was killed in the battle of Kolomna (1238).

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In anticipation of the "But could ancient Romans or Greeks defeat Mongols?" question I'll answer straight away: we don't know, they never faced each other, and they also were disconnected by hundreds of years of military evolution so it's unknown how Roman Empire could fare against Mongols were it to survive. But they surely were much more organized than later feudal states.

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Sources:

1. Roman Khrapachevsky, "Mongol army during the period of conquest of Ancient Russia", 2011.
2. Roman Khrapachevsky, "Military state of Chingis Khan", 2005.