I'm about as organic as possible, but I do occasionally experiment. I'm going to suggest using RoundUp but not as a spray. Yes I realize you want to be organic, but this is going to be only one, highly targeted, treatment that should really have no effect on anything but the weed. By getting rid of the weed once and for all, you can begin to enjoy your lawn sooner.

Start with super concentrated RoundUp. I think it's something like 43% strength. Put a little of that into a small bud vase or long bottle. Then take one strand of the weed and poke it down into the RoundUp. Prop up the vase or bottle so it doesn't spill out and let it sit for several weeks. I've used this approach to kill bindweed and it was a miracle. I put one strand of bindweed into the vase and after several weeks of seemingly not doing anything, SUDDENLY the entire plant was dead for a 30-foot diameter circle. It was incredible and nothing else was bothered.

You know in your case you might be able to paint one strand of the weed with RoundUp and get the same effect. That approach really minimizes the scary chemical fears that loom in the minds of us organic yard folks.

I have some more thoughts.

If the best tool against weeds was a healthy lawn, what's wrong with your lawn? Looking past the weed in the picture to the lawn, it's looking a little weak.

Where do you live?
What kind of grass do you have?
How have you been watering? How often and for how long? --and THIS is the key to having a healthy lawn.
When were the last two times you fertilized and what did you use?

Thanks dchall, I appreciate the input and the Roundup experiment sounds interesting and promising.

This October is two years for me in this house, so I still don't know much about the lawn as the house is about 45 years old. I only really paid attention to the lawn this past summer, primarily focusing on improving the soil structure for an overseed this fall. From what neighbors have told me, I believe that the house was fertilized synthetically for several years and then neglected for 2-3 years before we moved in.

We live in Central VA, zone 7.
Grass type is a mystery and a menagerie-I think I've identified ryegrass, bluegrass, and bentgrass in the existing lawn. I just spread a blend of 95% various tall fescues, 5% Spitfire "Thermal" Bluegrass at 5 lbs./1,000sf on Labor Day.
I have not watered the lawn consistently because I've been working on the soil more than the lawn in preparation to seed this fall. I plan to water 1 inch, once per week once this seed establishes.
I fertilized with alfalfa pellets at 20lbs/1000sf in May this year, and fertilized with EarthWorks Replenish 3-4-3 (http://soilfirst.com/replenish-3-4-3/) at 10lbs/1000sf immediately after seeding (two weeks ago). I also sprayed compost tea in May, June, and also two weeks ago.

The seed has germinated pretty well-I've been running around like a crazy person keeping the seed moist for the past two weeks. I have identified that it's been slower to germinate in the areas infested with the Japanese clover, and that these areas also generally correspond with the sunniest, hottest parts of the lawn. The section of lawn in the picture is a high-traffic area and I have a ways to go to get it up to snuff.

Short of the chemistry improvements you might make following a Logan Labs soil test, I'd say your soil is ready to go. The 2-year hiatus from chemicals was good. Then applying the organics was great. Now...just add water. Getting a deep watering routine going is the last thing. That could have been the first thing, but it needs to be a thing or else you won't have any lawn.

Just a note about your black asphalt: the black asphalt absorbs sunlight all day long and returns that energy to the air and soil in the form of heat all day and night. You'll need to water a little deeper adjacent to the paving. That's true of concrete and stone, but more so with black asphalt.

Great info, thanks. I'll see what happens to the clover as the lawn develops, and will try out the Roundup experiment if necessary.