We introduce spectral Morse-Smale analysis as a robust method to identify topological phase transitions in disordered continuous media. Combining microfluidic experiments with large-scale, pore-resolved simulations of porous media flow, we demonstrate that invariants of Morse-Smale graphs of flow speed provide a well-defined measure of the effects of spatial disorder on fluid transport. By systematically perturbing a microfluidic lattice, the fluid flow topology undergoes a phase transition from periodic to filamentous flow structure, which corresponds to a change in the spectral density of the Morse-Smale graphs and carries important implications for advective transport and front dispersion. Due to its generic formulation, spectral Morse-Smale analysis can be applied to detect and characterize topological transformations in a wide range of complex physical, chemical or biological fluid systems.