Competitive exclusion is frequently observed in the chemostat when multiple species compete exploitatively for a single growth-limiting nutrient. However, rigorous mathematical proof for competitive exclusion is very limited, especially for delayed chemostat models with differential removal rates. In this paper, by employing the method of Liapunov functionals and placing two assumptions on the response functions, we prove that competitive exclusion holds under a generic condition for chemostat models with differential removal rates and discrete (also finitely distributed) delays. For delayed chemostat models with a large variety of widely used response functions, including Holling types I, II, and III and even some nonmonotone response functions, our results show that the competition outcome is completely determined by the species' break-even concentrations: it is the species with the lowest break-even concentration that survives in the chemostat and drives other species to extinction.