The initial transient phase of an emerging epidemic is of critical importance for data-driven model building, model-based prediction of the epidemic trend, and articulation of control/prevention strategies. In principle, quantitative models for real-world epidemics need to be memory-dependent or non-Markovian, but this presents difficulties for data collection, parameter estimation, computation and analyses. In contrast, the difficulties do not arise in the traditional Markovian models. To uncover the conditions under which Markovian and non-Markovian models are equivalent for transient epidemic dynamics is outstanding and of significant current interest. We develop a comprehensive computational and analytic framework to establish that the transient-state equivalence holds when the average generation time matches and average removal time, resulting in minimal Markovian estimation errors in the basic reproduction number, epidemic forecasting, and evaluation of control strategy. Strikingly, the errors depend on the generation-to-removal time ratio but not on the specific values and distributions of these times, and this universality will further facilitate prediction rectification. Overall, our study provides a general criterion for modeling memory-dependent processes using the Markovian frameworks.