Yes. Dogs do sweat, but only on the hairless pads of their feet.

However, their major method of temperature control is through panting, usually with their tongues dangling outside of their mouths. This allows transfer of heat from their core body/chest area straight out of their mouths through their breath. In this method they don't lose as much fluids as people do when they cool by perspiration, since, to cool through perspiration with evaporation, you have to create a much larger wet area to provide cooling of the skin and that loses more body fluids. It is much less efficient than the method dogs use, and it also often creates dehydration in people.