The only way to eliminate Johnson grass, Quackgrass, Sheep Sorrel, or other "weeds" that spread by rhizomes is to dig out all of the rhizomes and as soon as you see any new growth pull that out, trying very hard to get all of the root (rhizome) at the same time. People spray all kinds of material in an attempt to kill this stuff, and the misleading ads kind of give the impression these poison will eliminate these "weeds", but they do not. If those products actually did what the ads said they did you would not need to buy more each year.

Johnson grass can be controlled by repeated, diligent removal of above ground plant parts. Details in the link. Glyphosate based herbicides such as Roundup will control Johnson and other perennial rooted grasses effectively, although it may take several applications. Apply glyphosate when Johnsongrass is actively growing and between 12 and 24 inches tall. Of course applications of this material will do nothing to prevent the future re-invasion of these grasses, either by seed or by rhizomes from adjacent weed patches. Glyphosate is not herbicidally active once it hits soil, and it will break down completely over time.

Once your target weeds are controlled, make sure to remove any future weed seedlings and apply a protective mulch.

Here is a link that might be useful: Attra link

Glyphosate (roundup) will kill all of your adjoining plants and it will stay active in the soil. If you use it in your Garden it will poison the soil. I have lived through this nightmare and have been an active gardener for 30+ years. No matter what the proponents of Monsanto claim.

Smother if you don't want to dig.