When a dog urinates on the floor, the usual solution is to wipe it off with paper towel or tissue.
But I find this more useful: female sanitary pads. It's a lot more absorbent, and won't seep through to our hands so they won't get dirty.
There are some very cheap brands in the market, so they won't hurt our pockets too much. And in most cases, the smaller sizes without wings will work just fine.