Cynodon dactylon

Aggressively invades yards, gardens, and riparian areas; inhibits the natural meandering of streams; competes with native plants for space, light and nutrients, and forms dense mats that inhibit the growth of other species. Removal is difficult because of an extensive system of underground stems, up to 1 ft deep in soil. Bermuda grass can also be a hay contaminate.

Turf and lawn grasses are often invasive, require regular maintenance and are high water users. If closely planted and mowed, the two native grasses recommended below will resemble turf.
Blue grama Bouteloua gracilis
A low-growing perennial bunch grass up to 1.5 ft tall with blue-green foliage. Seed heads appear in summer, curling when dry to look like eyelashes. Grama grass grows into an attractive specimen or turf. Wildlife relish this palatable and nutritious grass. Blue grama survives heat, drought, and cold.
Curly mesquite Hilaria belangeri
A drought tolerant sod-forming grass which spreads by above-ground runners, growing up to 12 in tall. It may be hard to find, try nurseries specializing in native plants.

Other features to consider instead of a grass lawn:
Rockscape, cactus garden, native wildflower garden, gravel, sand, or artificial turf.