We're dealing with the same type of thing in our yard. Spraying it with Roundup and then keeping it covered with black plastic for a season or two helped, though it didn't get rid of it completely. You might try that if it's in a hidden area. If it's in your front yard (as ours now is), I don't know what to do. I've hard that several seasons of Roundup in a row will eventually poison all the rhizomes, but who wants to poison their lawn for two years straight?

Looks like my next door neighbors centipede! I go out every winter now when it's dormant and pull the runners out. Read where 2-4D will kill it but it also killed my Bermuda. I used to think Bermuda was pretty invasive but this Centipede is much worse IMO.

There is a reason why it looks like the golf course. Your description of the growth habits and the pictures (while not perfect specimens) make me think that you may be dealing with Bentgrass.

What is the desirable grass that the Bentgrass is invading? We need that answer to help select a solution.

The existing grass is "mostly" KBG".