I appreciate your photos, especially the close-ups of the leaves which made identification much easier. The weed found in your lawn are most likely Heartleaf Drymary, Dyrmaria cordata, also called West Indian Chickweed. This weed is weak-stemmed which means it breaks easily and therefore the seeds can effortlessly spread to other areas. It is an annual and therefore can only reproduce by seed.
The seeds and fruit will stick to shoes, socks and clothing as well as pet fur and distribute to other areas of the yard. Heartleaf Drymary prefers moist to wet soils and this can mean the area where this weed is located might be receiving too much water or the area is not well drained. Heartleaf Drymary can be found in all parts of Georgia, Louisiana and Florida. India and Sri Lanka use it as erosion control, especially on tea plantations. Heartleaf Drymary is also commonly found under coffee trees in Ethiopia and East Africa. It does have one redeeming quality – the leaves are edible and can be used in salads.
Best method of control is to use a pre-emergent in fall and spring. You should also check the amount of water the grass is receiving in this area. Remember, apply ½ to ¾ of an inch of water to the lawn if it is in full sun, shady areas receive less water and fertilizer. St. Augustinegrass prefers to be watered deeply but less often. Shallow, frequent irrigation produces short roots. Long roots (4-6 inches) help grass survive. Hand pulling weeds is always an option, especially for annuals and the sooner the better. As soon as the plant starts producing flowers and seeds, it is too late for chemical control.