Transient phenomena are observed across natural systems. Their study holds the key to understand issues from climate change to the ecological impact of invasive species and antibiotic resistant bacteria. Contrary to equilibrium dynamics, however, there is no systematic methodology to study transient dynamics. Here we present a new framework to address this. The approach transforms the dynamical system into dimensionless logarithmic variables. This facilitates the application of an order of magnitude analysis that separates the dynamics into regions that can be solved using approximate analytical expressions. The effectiveness of this method was assessed for the transient dynamics of E. coli bacteria and T4 phage for different initial conditions. The analytical expressions unveiled how the different phage and bacteria traits contributed to the dynamics, and the analytical results had a median accuracy of 98% with respect the exact numerical solutions. The method was also applied to obtain alternative values for the phage-bacteria interaction traits that could generate reduced transient dynamic scenarios. The generality and accuracy of the approach will facilitate a systematic study of transient dynamics in natural systems.