Context: Management of suites of invasive mammal species can lead to perverse outcomes, such as meso-predator release, or can achieve desirable reductions in the abundance of top-order predators by controlling their prey. Predictive models for predator–prey systems require estimates of predator functional responses, i.e. predation rates as functions of prey density. Aims: In New Zealand, estimates of the functional responses of stoats (Mustela erminea) to mice (Mus musculus) and ship (black) rats (Rattus rattus) are required to improve management models for these invasive species. Methods: We derived fitted relationships between the presence or absence of mouse or ship-rat remains in stoat guts and corresponding indices of prey abundance in beech and podocarp forests, respectively. To convert field data on stoat-gut contents to minimum kill rates, we used data on feeding activity and estimates of gut-passage time, observed in captive stoats. Key results: The most parsimonious fitted curves were Type II functional responses, with a steeper stoat–mouse curve for autumn–winter, indicating a more specialist feeding habit than that in spring–summer. Estimated kill rates of mice per stoat per day reached an asymptote of 1.13 during autumn–winter. Our maximum observed kill rate for spring–summer was 11% less than the extrapolated upper limit of 1.04 mice per stoat per day for New Zealand ecosystems. No asymptote was reached within the limits of the data for the stoat–rat relationship. Conclusions: Recent models for trophic interactions between stoats and the primary rodent prey have overestimated kill rates by stoats in forested ecosystems, particularly at very low and very high densities of mice. We show how data on stoat-gut contents can be rescaled to estimate minimum kill rates of rodent prey. Implications: The functional-response relationships we have derived can be used to improve modelled predictions of the effects of natural or management-driven perturbations of invasive stoats and their primary rodent-prey populations.