Mechanically destroy the tree and remove its root system. Cut the mesquite tree close to its base with a chainsaw. Use a plow or a tractor-pulled root grub to pull up the roots.
Mix a commercially available triclopyr ester herbicide designed for range brush control at a 15 percent ratio with diesel oil or vegetable oil. Treat cut stumps that are not grubbed with the mixture to kill the buds and stems at the base of the stump. Apply at any time of the year.
Spray the leaves of standing trees with a mixture of 0.5 percent triclopyr ester herbicide, 0.5 percent clopyralid herbicide and 0.25 percent surfactant mixed with water. Spray during the late spring when the soil temperature is at least 75 degrees Fahrenheit and the trees are dark green. It may be helpful to add an ounce of dye per gallon of herbicide to readily identify which trees have been treated.
Apply an hexazinone herbicide to the soil with a precision delivery handsprayer. Spray the hexazinone around the mesquite tree within three inches of the root color. Adjust the sprayer to deliver 2 to 4 milliliters of solution per inch of tree stem diameter at breast height. Avoid treating wet soil or when rain is forecast.
Use prescribed burning sparingly. Fire will not completely eliminate mesquite because the subterranean root structure will survive. However it may be used to top-kill young mesquite and may supress regrowth if the burns are repeated at intervals of every three to five years.