We report the discovery of superconductivity at 2.3 K in Zr5Sb3, the first superconducting member in the large compound family of the Mn5Si3-structure type. Transport, magnetic, and calorimetric measurements and band structure calculations show it to be a phonon-mediated BCS superconductor, with a relatively large density of states at the Fermi level associated with the d-electrons of Zr and substantially larger electron-phonon coupling compared to the Sn counterpart compound Zr5Sn3. More superconductors with even higher transition temperatures are expected to be found in this family of compounds.