Those 1” moths are called Sod Webworm moths and they do not cause damage to your lawn. They fly low to the ground when disturbed and hide in sheltered areas, such as St. Augustine leaf blades, and shrubs during the day. There is no control for the adult moth.

The caterpillar of this moth does damage your lawn, but only if the adult moth lays eggs in your yard. The gray-green ¾” larvae will chew on grass blades in your lawn, causing it to look ragged and mowed closely in patches. This pest is most active from April through November. Look closely for chewed leaves, silken webs, green droppings and gray-green ¾” spotted caterpillars curled up on their sides.

Spray the affected area with a recommended insecticide such as Sevin or Dylox, per label instructions. A non-chemical option for control is Bacillus thuringiensis. . Do not spray the flying moths as this will be ineffective. 

Additional Resources

http://edis.ifas.ufl.edu/pdffiles/IN/IN60800.pdf

http://edis.ifas.ufl.edu/pdffiles/IN/IN02500.pdf