It sounds like you might have creeping Charlie…a vine lawn weed. You’ll need something stronger like Bonide Chickweed and Clover Killer. It can be used on lawn and won’t kill the grass when used in accordance with the directions on the label. Now through mid-September is a great time for lawn weed killing. Here’s a link to the product description.
http://www.bonide.com/products/weed-control/view/0612/chickweed-clover-and-oxalis-killer-rtu