Ask Mr. Smarty Plants

This puts us in a slightly embarrassing position. The Lady Bird Johnson Wildflower Center is dedicated to the use, protection and propagation of plants native not only to North America but to the area in which it is being grown. Your "weed," Hieracium venosum (rattlesnakeweed) is native to South Carolina, while your lawn, Ischaemum muticum, Centipede Grass, is native to China and parts of southeast Asia. Turns out both plants are considered by some to be invasive in nature. One source said that centipede grass can become a weed in many annual and perennial crops and can develop into huge thickets in drainage canals and ditches. 

However, to you the rattlesnake weed is the invasive. We have no personal experience with it (but we do with rattlesnakes!) but from the information we have it looks pretty hard to pull out, which is always our first choice in getting rid of something like that. Preventing it from blooming, so it cannot set more seed, is the first line of defense. As those blooms stand up pretty high off the ground, getting them mowed early will certainly help. 

Much as we hate to prescribe herbicides, because of environmental reasons, that may be your last resort. Centipede grass is a monocot, like all grasses. Rattlesnake weed is a dicot, also classified as a broad-leaf plant. You can buy herbicides specifically for either; obviously, you want one for a dicot, or broad-leaf plant. Read the instructions carefully, and follow them closely. Broadcast spraying could, with a gust of wind, damage some shrubs, flowering plants or trees, because they are dicots, also. 

Conclusion: Prevent the Hieracium venosum (rattlesnakeweed) from seeding if you can, pull it out of the ground when it's possible, use a dicot herbicide as a last resort. But mind the rattlesnakes!

From our Native Plant Image Gallery:

Hieracium venosum

Hieracium venosum

Hieracium venosum