Recently, A2B3 type strong spin orbital coupling compounds such as Bi2Te3, Bi2Se3 and Sb2Te3 were theoretically predicated to be topological insulators and demonstrated through experimental efforts. The counterpart compound Sb2Se3 on the other hand was found to be topological trivial, but further theoretical studies indicated that the pressure might induce Sb2Se3 into a topological nontrivial state. Here, we report on the discovery of superconductivity in Sb2Se3 single crystal induced via pressure. Our experiments indicated that Sb2Se3 became superconductive at high pressures above 10 GPa proceeded by a pressure induced insulator to metal like transition at 3 Pa which should be related to the topological quantum transition. The superconducting transition temperature (TC) increased to around 8.0 K with pressure up to 40 GPa while it keeps ambient structure. High pressure Raman revealed that new modes appeared around 10 GPa and 20 GPa, respectively, which correspond to occurrence of superconductivity and to the change of TC slop as the function of high pressure in conjunction with the evolutions of structural parameters at high pressures.