This study focuses on the development of an online inertial mirror descent algorithm for characterizing a novel class of averaged ε-Nash equilibrium in a class of non-cooperative multiplayer games with dynamic strategies in continuous time. Each player’s dynamic strategy is subject to constraints defined on a compact and convex set. Using the Tanaka-Yokohama formula, which characterizes the Nash ε-equilibrium, the strategy is determined for each player in the dynamic game. The Min–Max property of this function confirms the existence of the Nash ε-equilibrium. The algorithm design employs the Legendre–Fenchel transform and a selected proxy function to facilitate the inertial mirror descent approach for the averaged trajectories of the game dynamics. This transformation is instrumental in proving the convergence to the Nash ε-equilibrium with a rate of O(t−1). In addition, various proxy functions are proposed and analyzed for their effectiveness in constructing the online inertial mirror descent algorithm. A numerical example contributes to evidence of the application of the mirror descent algorithm presented in this study, considering two players with a state of three components each. A particular selection of a proxy function characterizes the existence of the Nash ε-equilibrium.