Surprisingly, the mongols were not the undisputed masters of the battlefield. They were good, and I will get into that latter, but it is important to understand that they suffered defeat on the battlefield many times. The thing that made them so successful was their unprecedented ability to capitalize on their successes while preventing their enemies from capitalizing on the mongols defeats. That is to say, they had a strategical advantage.

If the mongols win a battle it will be an almost total victory. Maybe some of the Calvary can get away but any infantry is going to be surrounded by the much faster and more mobile Mongol forces and get captured or killed. Conversely if the Mongols lose a battle their superior speed will allow them escape relatively unharmed. Maybe you could send your cavalry after them, but if you send them too far away from the infantry then the mongols might turn and encircle your now exposed cavalry. This is compound by the fact that the mongol armies were very self sufficient, meaning they didn't have to run away in a particular direction in order to meet up with supplies so the opposing commander can't predict where they will show up.

While their tactics were not amazing, they were good enough. Good enough for their strategical and logistical advantage to make up for it. They relied heavily on Horse Archers and Horse Archers are amazingly powerful on the battlefield. However most armies that used Horse Archers didn't use them to nearly the same extent for a simple reason, wet bows cannot be used (It messes with the bowstring), so you can't use Horse Archers when its raining (most armies that used Horse Archers trained them as hybrid cavalry for this very reason). However the Mongols superior logistical ability and maneuverability meant that they could just refuse to fight when it was raining and just avoid the opposing army until they could fight on their terms (This was one of the reasons they had so much trouble invading India, it was just too wet too much of the time).

Now when they encountered something they had no experience with, they paid people to teach them how to deal with it. One of things that shocked the Chinese was how quickly the Mongols were able learn to conduct a successful siege assault and this was due to expertise the mongols obtained from foreign specialists (this really does show how the Mongols were in no way dumb brutes, they saw the importance of obtaining expertise in areas they were inexperienced with and they absorbed that education like a sponge).