Grassland-based animal husbandry is the dominant economic activity on the Qinghai-Tibetan Plateau, and livestock production is the main contributor to livelihood creation for pastoralists. In recent years, changes in food markets and climate variables bring the need to develop livestock production into sharp focus. By using the Cobb–Douglas production function, we defined the per capita meat production of rural households as dependent variable, while the proportion of accumulated precipitation from April to September in annual average precipitation (PCI), the productive fixed assets per capita, the number of labor force per household were considered as independent variables in order to model the relationship between the per capita meat production and the PCI at micro level of household. Based on the time series data of rural household survey (1986–2012) and observed meteorological data (1978–2010) across Shigatse Prefecture, Tibet, China, we estimated the sensitivity of per capita meat production to the PCI. We also quantified the adaptation effects of the productive fixed assets and labor productivity in improving per capita meat production of rural households respectively. Our results indicate that: (i) the per capita meat production of rural households exhibits a high sensitivity to three key variables: the productive fixed assets per capita, the number of labor force per household, and the PCI. It increases by 0.938 and 0.218 times when the productive fixed assets per capita and the PCI increase by one unit respectively. On the contrary, it decreases by 0.268 times when the number of labor force per household engaged in grass-fed livestock sector increases by one unit; (ii) on the basis of parameter estimates and Cobb–Douglas modeling, the losses of meat production due to decrease in the PCI can be offset by increasing productive fixed assets. And a decrease in the labor force engaged in grass-fed livestock sector can help improve the productivity of grassland-based animal husbandry. Therefore, adaptation options of rural households, based on the productive fixed assets and labor productivity, will be most effective in the long term when they are complemented by appropriate public policies such as infrastructure construction, water conservancy project, subsidies of livestock production material, professional training and innovation of pastoralists’ perception at different scales.