Low energy reaction dynamics can strongly depend on the internal structure of the reactants. The role of rotationally inelastic processes in cold collisions involving polyatomic molecules has not been explored so far. Here we address this problem performing a merged-beam study of the He*+CHF$_3$ Penning ionization reaction in a range of collision energies $E/k_B$=0.5--120 K. The experimental cross sections are compared with total reaction cross sections calculated within the framework of the quantum defect theory. We find that the broad range of collision energies combined with the relatively small rotational constants of \chfs makes rotationally inelastic collisions a crucial player in the total reaction dynamics. Quantitative agreement between theory and experiment is only obtained if the energy-dependent probability for rotational excitation is included in the calculations, in stark contrast to previous experiments where classical scaling laws were able to describe the results.