Since it is a native plant, it is NOT, by definition, considered invasive. But an aggressive spreader for sure. You need to remove the root system if you want to remove the plant, as it spreads by underground rhizomes.

The leaves can look a bit like dayflower. Here's a link that might help. When it mentions a "ligule" it means a membranous bit of tissue on the inside of the leaf where it meets the stem. Sometimes the ligule can have hairs in panic grasses. Here is a link that might be useful: Deer-tongue Panic Grass

It's one of half dozen or so speedwell (Veronica spp) that are common lawn weeds. (Or not, depending on how you look at it). They should respond to any broadleaf lawn weed killer.

Thanks for identifying this plant. You seem to be right! I've started treating it and it appears to be killing it off.