This paper deals with the state estimation of non-linear and non-Gaussian systems with an emphasis on the numerical solution to the Bayesian recursive relations. In particular, this paper builds upon the Lagrangian grid-based filter (GbF) recently-developed for linear systems and extends it for systems with nonlinear dynamics that are invertible. The proposed nonlinear Lagrangian GbF reduces the computational complexity of the standard GbFs from quadratic to log-linear, while preserving all the strengths of the original GbF such as robustness, accuracy, and deterministic behaviour. The proposed filter is compared with the particle filter in several numerical studies using the publicly available MATLAB\textregistered\ implementation\footnote{https://github.com/pesslovany/Matlab-LagrangianPMF}.