Mow the elephant grass as close to the ground as you can with a lawn mower or brush hog once the grass surpasses its growing season. Ideally, this is after the first frost. Cutting the grass after the growing season will prevent grass seed from becoming airborne and seeding surrounding areas.
Rake and remove all the cut grass from the area.
Apply an herbicide to the freshly mowed grass to kill it. There are many commercial herbicides on the market. Follow the dosing recommendations on the packaging to maximize the effect of the herbicide.
Till the area, after the grass turns brown, with a motorized tiller to eradicate the root system of the elephant grass. Repeat tilling after the spring thaw.
Plant grass seed in the affected area to prevent the elephant grass seed from coming back.