We observe the suppression of inelastic dipolar scattering in ultracold Fermi gases of the highly magnetic atom dysprosium: the more energy that is released, the less frequently these exothermic reactions take place, and only quantum spin statistics can explain this counterintuitive effect. Inelastic dipolar scattering in nonzero magnetic fields leads to heating or to loss of the trapped population, both detrimental to experiments intended to study quantum many-body physics with strongly dipolar gases. Fermi statistics, however, is predicted to lead to a kinematic suppression of these harmful reactions. Indeed, we observe a 120-fold suppression of dipolar relaxation in fermionic versus bosonic Dy, as expected from theory describing universal inelastic dipolar scattering, though never before experimentally confirmed. Similarly, low inelastic cross sections are observed in spin mixtures, also with striking correspondence to predictions. The suppression of relaxation opens the possibility of employing fermionic dipolar species in studies of quantum many-body physics involving, e.g., synthetic gauge fields and pairing.