For drug safety analysis in clinical trials where adverse events are often rarely observed, a challenging but recurrent problem is the present of the so-called zero total event studies where both treatment and control groups observe zero events. In this situation, conventional meta-analysis approaches either exclude such studies from the analysis, or apply 0.5 continuity corrections to zero events. Both practices, however, are known to have undesirable consequences in inference. Motivated by this problem, we propose to combine p-value functions (also known as significance functions) associated with the exact tests from individual studies. This approach can incorporate all available data in the analysis without using artificial corrections for zero events. We further show that the idea of combining p-value functions yields a unifying framework for exact meta-analysis, as it permits broad choices for the combining elements, such as tests used in individual studies, and any parameter of interest. We theoretically show that our approach yields statements that explicitly account for the impact of individual studies on the overall inference in terms of efficiency/power and the type I error rate. We demonstrate, through numerically studies in some rare events settings, that our exact approach is efficient and outperforms existing commonly used meta-analysis methods.