Black-eyed Susans can be grown as annuals if you take precautions as the flowers start to seed. Cutting the flowers before they dry and wither means that the thousands of tiny seeds stay attached to the flower head. If new plants sprout the next spring, dig them out by hand.
Black-eyed Susans are broadcast seeders, meaning their seeds are carried to ground by wind, rain, bees, birds and other factors. Stopping the seeds from dropping as the flowers die prevents black-eyed Susans from growing in your garden again. Digging them up also keeps the plants from sprouting again the next year.
Fortunately, black-eyed Susans are very drought tolerant, needing very little water to stay healthy and bloom. If you’re growing them for only one season, water as little as possible, so that you can tell when the flowers start to wilt. While they’re still in good shape, cut them for flower arrangements. This eliminates most of the seeds.
If you want to keep the seeds for later use or to share with other gardeners, cover each flower head as blooms mature with a plastic bag and close it with a twist tie. Then cut the flower. This catches all the seeds as well as packages them for easy storage.