Never ever use autoresolve against the Mongols if the army contains their original troops. The autoresolve function is basically a number crunch. It's apparently heavily weighted in favor of numbers and experience. The first is usually a disadvantage for cavalry, as mounted units are smaller in size, but all the original Mongol invasion troops have something like 6 xp! The result is that you will almost always be crushed if you use the computer to resolve it.

Now, if you cannot fight the battles out yourself for whatever reason, then I have to agree with RVG. Build your armies as full stacks of big infantry units like spear militia or Byzantine Spearmen, keep multiple stacks together, and then resolve it. You'll take horrific losses, but those troops are pretty cheap to replace, and you at least have a decent chance of winning.

If you can fight out the battles, then the Byzantines can give the Mongols a serious challenge on the field. I've beaten them in an open field battle with an army of 2 generals, 2 kataphractoi, and 16 assorted horse archers, but there's no way to fight that battle without losing a lot of troops. It works out better if defending a bridge or a ford.