In this Letter, we report the first measurement of the inelastic cross section for antideuteron-nucleus interactions at low particle momenta, covering a range of 0.3≤p<4 GeV/c. The measurement is carried out using p-Pb collisions at a center-of-mass energy per nucleon-nucleon pair of sNN=5.02 TeV, recorded with the ALICE detector at the CERN LHC and utilizing the detector material as an absorber for antideuterons and antiprotons. The extracted raw primary antiparticle-to-particle ratios are compared to the results from detailed ALICE simulations based on the geant4 toolkit for the propagation of (anti)particles through the detector material. The analysis of the raw primary (anti)proton spectra serves as a benchmark for this study, since their hadronic interaction cross sections are well constrained experimentally. The first measurement of the inelastic cross section for antideuteron-nucleus interactions averaged over the ALICE detector material with atomic mass numbers A