Earth System Models (ESMs) are essential tools for understanding the impact of human actions on Earth's climate. One key application of these models is studying extreme weather events, such as heat waves or dry spells, which have significant socioeconomic and environmental consequences. However, the computational demands of running a sufficient number of simulations to analyze the risks are often prohibitive. In this paper we demonstrate that diffusion models -- a class of generative deep learning models -- can effectively emulate the spatio-temporal trends of ESMs under previously unseen climate scenarios, while only requiring a small fraction of the computational resources. We present a diffusion model that is conditioned on monthly averages of temperature or precipitation on a $96 \times 96$ global grid, and produces daily values that are both realistic and consistent with those averages. Our results show that the output from our diffusion model closely matches the spatio-temporal behavior of the ESM it emulates in terms of the frequency of phenomena such as heat waves, dry spells, or rainfall intensity.