Dig individual marsh marigolds as they emerge from the lawn. Push any disturbed grass clumps back into place, and water well to reduce the stress caused by damaging grass roots. Dig out large clumps of marsh marigolds if you intend to replant the lawn anyway, and manage the stragglers as they appear.
Mow your lawn weekly, making sure to cut the marsh marigolds as close to the ground as you can without damaging your lawn. Monitor the problem closely, you don't want to allow the marigolds to set flowers or put out too many leaves -- flowers lead to more plants and leaves will allow the plant to feed itself while you're trying to starve it out. Continue mowing frequently throughout the growing season.
Clip marsh marigold flowers and leaves off of the plants between mowings if your infestation is small, but aggressive. Back off on supplemental lawn watering to stress the marsh marigolds, but don't allow your lawn to die trying to dry the nuisance plants out -- if your lawn looks brown or limp, water regardless of the marsh marigolds.
Fashion a cardboard collar for the marsh marigolds that will protect your lawn from overspray and apply a pre-mixed glyphosate herbicide to the leaves of the marigolds in the fall, once temperatures are between 65 and 80 degrees Fahrenheit. Repeat treatments every 20 to 30 days, or until temperatures are below 60 degrees -- foliar herbicides work best on plants that are actively growing.