Pull the desirable plants and pot them in temporary pots. Then buy black plastic and cover the area until all of the plants are dead. After you uncover the area in several weeks, hold off on replanting your desired plants, until you know that the problem plant has been eliminated.
Someone mentioned that the plants may be annuals. If that is the case, then you could leave your perennials in place and simply use a pre-emergent herbicide such as treflan prior to the next growing season. Pre-emegent herbicides will kill any seeds in the soil but won't bother any existing plants.
Eclectic gardening style, drawing from 45 years of interest and experience. Mostly plant in raised beds and containers primarily using intensive gardening techniques.