It's likely either bermudagrass or zoysiagrass, both of which can be quite invasive. There is a grass killer which goes by names like Grass B Gon, or Over-the-top grass killer, which kills grasses but leaves (most) ornamentals unharmed. That's the least labor intensive way to do this. (Be sure to read the label. There are a few ornamentals that it will harm.)
Nonchemical control is more difficult. You first to have to eliminate the grass that's already invaded, which requires a LOT of digging and careful removal of all the rhizomes. Then you can put down weed fabric and mulch on top of that and that should help keep it out.
Either way, some sort of edging product between the lawn and flowerbed will help keep the grass from re-invading. Otherwise you should spray herbicide along the edge of the flowerbed ("chemical edging" they call it) every month or so to keep the grass out.