Classic consumer-resource models with hyperbolic functional responses predict that enrichment increases the average biomasses of the species, but eventually leads to species’ extinction due to accelerated oscillations (“paradox of enrichment”). However, empirical studies have stressed the complexity of natural food webs and the dominance of sigmoid or predator-interference functional responses, which may dampen population oscillations due to enrichment. Using analytical and numerical methods, we study enrichment effects on simple consumer-resource pairs and complex food webs with hyperbolic Holling type II (hereafter: type II), sigmoid Holling type III (hereafter: type III) and Beddington-De Angelis predator-interference functional responses (hereafter: BDA). Consumer-resource systems with a type III or BDA functional response are highly robust against accelerated oscillations due to enrichment, and the “paradox of enrichment” is resolved under certain parameter combinations. Subsequently, we simulated complex food webs with empirically-corroborated body-size structures of consumers that are ten times larger than their average resource. Our analyses demonstrate positive connectance-stability relationships with BDA or type III functional responses. Moreover, increasing connectance of these food webs also increases the robustness against enrichment in models with a BDA functional response. These results suggest that the well-known destabilising effects of connectance and enrichment found in classic models with type II functional responses may be inverted into stabilising effects in more realistic food-web models that are based on empirically-corroborated body-size structures and BDA or type III functional responses.