The motion of droplets on typical surfaces is generally halted by contact line pinning associated with contact angle hysteresis. In this study, it was shown that, on a zwitterionic sulfobetaine silane (SBSi)-coated surface, aqueous drops with appropriate solutes can demonstrate hysteresis-free behavior, whereas a pure water drop shows spontaneous spreading. By adding solutes such as polyethylene glycol, 2(2-butoxy ethoxy) ethanol, or sodium n-dodecyl sulfate, an aqueous drop with a small contact angle (disappearance of spontaneous spreading) was formed on SBSi surfaces. The initial drop shape was readily relaxed back to a circular shape (hysteresis-free behavior), even upon severe disturbances. Moreover, it was interesting to observe the self-propulsion of such a drop on horizontal SBSi surfaces in the absence of externally provided stimuli. The self-propelled drop tends to follow a random trajectory, and the continuous movement can last for at least 10 min. This self-propelled random motion can be attributed to the combined effects of the hysteresis-free surface and the Marangoni stress. The former comes from the total wetting property of the surface, while the latter originates from surface tension gradient due to fluctuating evaporation rates along the drop border.