Snip off any sucker growth with pruning shears.
Notch the surface of the salt cedar trunk with a knife or axe near the bark if it is an older stump, and not a freshly cut tree. The cuts opens the surface to allow the chemical to soak in more quickly. The material will need to enter the tree's vascular system, which is located in the inner bark of the tree.
Pour a pre-mixed herbicide, such as glyphosate, into a small container with a wide mouth. Glyphosate kills by working through the stalk and root system of a plant, but becomes inert when it comes in contact with the soil. It won't cross from one root system to another and kill your grass. According to the Department of Ecology of the State of Washington, both triclopyr and glyphosate are effective in killing salt cedar stumps, eliminating sprout issues. Triclopyr is more effective in killing stumps, but can be absorbed by roots through the ground, so it should be used cautiously to prevent any spills. Dilute triclopyr before use adding 1 part triclopyr to 1 part water.
Dip a brush into the wide-mouthed container and brush the chemical onto the stump, making sure to coat near the inner edge of the bark . Applying glyphosate in this manner prevents you from spraying the surrounding lawn area, and killing it.
Repeat the process each time new growth appears until suckers no longer develop from the trunk or roots.