I had a patch of crab grass in middle of lawn. Because it expands by its roots and produces an unbelievable number (millions) of seeds per ounce, this is how I got rid of it:

1. I dug out the sod in a 3 foot radius around the crab grass.

2. I put the sod into the green waste bin for pickup by the waste management truck every week until it was all gone (a little at a time as it is quite heavy with all the dirt that goes with the sod).

3. I left the patch bare for 2 years. One piece of crab grass showed up so I also threw it out.

4. Finally no more crabgrass. It didn't come back (time to re-sod).

5. Optional step: I thought about cooking the soil with a propane torch after I removed the sod to kill any remaining seeds. Seemed like too much work, but if you really want to be sure ...