They used saddles. See for example this exhibit in the National Museum in Ulaanbaatar. Mongol warriors were able to shoot from horseback in part because they had stable saddles and short stirrups in which they could stand up.
The linked picture is not mine~~, I'll try to find a better one from my visit~~. It was well explained in the museum.
Edit: here are some of my own pictures from the same display: http://imgur.com/0kKcOuz http://imgur.com/4x5y34x
Mongolian warriors used saddles, as is described below. This is an example of a 13th century Mongolian saddle made of wood. I do not have much more on the subject of saddles, but I have written a couple of answers regarding Mongol military tactics, supply routes, rations, and other areas of interest.
Beneficial advances during the reign of Genghis Khan
Regarding Mongolian Steppe Ponies Horses, I've found a very good look at them from the Smithsonian National Zoo. It has a very interesting video over PRZEWALSKI'S HORSE, which was mentioned earlier.
EDIT: At the request of /u/Ubrrmensch, steppe ponies is now steppe horses.
