Experts and BHG readers answer.
Bermuda Grass in garden
Bermuda grass (Cynodon dactylon) is durable and rapid-spreading, and it makes for a nightmarish invasive plant in the landscape, just as you describe. It spreads three ways; stolons above ground, rhizomes below ground, and seeds. Plastic or metal landscape edging between your turf and garden beds won't keep it from invading, but you probably already knew that. Whenever you dig, you're stirring dormant seeds up to the surface. There are some grass-specific herbicides that should leave your broadleaf perennials and shrubs unharmed. They include herbicides with the active ingredients fluazifop or sethoxydim. Products include Grass Getter, Ornamec, and Grass-B-Gon, among others. Because Bermuda grass is deeply rooted and surviving segments rapidly re-root, you'll have to do repeated applications beginning in early spring and throughout the summer. Other control methods include creating more shade in the garden, repeated pulling, and late-summer applications of the non-selective herbicide Roundup, which kills everything it contacts.