This study proposes a novel approach, developing and analyzing a higher-order, structure-preserving discretization method for inviscid barotropic flows from a Lagrangian perspective. The discretization encodes flow variables as discrete differential forms on a space-time mesh, using principles from differential geometry and algebraic topology. Unlike standard Lagrangian methods, which are prone to mesh distortion, this framework computes fluid deformations in a reference configuration and systematically maps them to the physical domain. This structure-preserving design ensures that fundamental conservation laws for mass, momentum, and energy hold up to machine precision. It also efficiently handles low-Mach number flows without specialized fixes for stiff pressure waves. Numerical experiments on expansion and compression flows confirm the discretization’s accuracy, stability, and conservation properties. The formulation naturally couples with structural solvers, enabling fluid-structure interaction and other multi-physics applications. By uniting spectral accuracy with a geometry-aware design, this approach serves as a first step toward a complete Lagrangian spectral element solver.