The emergence of infectious diseases with pandemic potential is a major public health threat worldwide. The World Health Organization reports that about 60% of emerging infectious diseases are zoonoses, originating from spillover events. Although the mechanisms behind spillover events remain unclear, mathematical modeling offers a way to understand the intricate interactions among pathogens, wildlife, humans, and their shared environment. Aiming at gaining insights into the dynamics of spillover events and the outcome of an eventual disease outbreak in a population, we propose a continuous time stochastic modeling framework. This framework links the dynamics of animal reservoirs and human hosts to simulate cross-species disease transmission. We conduct a thorough analysis of the model followed by numerical experiments that explore various spillover scenarios. The results suggest that although most epidemic outbreaks caused by novel zoonotic pathogens do not persist in the human population, the rising number of spillover events can avoid long-lasting extinction and lead to unexpected large outbreaks. Hence, global efforts to reduce the impacts of emerging diseases should not only address post-emergence outbreak control but also need to prevent pandemics before they are established.