© 2019 American Physical Society. We systematically investigated the pressure dependence of electrical transport and the crystal structure of topological insulator, Bi2-xSbxTe3-ySey, which showed no superconductivity down to 2.0 K at ambient pressure. The Bi2-xSbxTe3-ySey crystal showed two structural phase transitions under pressure, from rhombohedral structure (space group No. 166, R3m, termed phase I) to monoclinic structure (space group No. 12, C2/m, termed phase II), and from phase II to another monoclinic structure (space group No. 12, C2/m, termed phase III). Superconductivity appeared when applying pressure; actually the superconductivity of all Bi2-xSbxTe3-ySey samples emerged in phase I. The superconducting transition temperature, Tc, increased against pressure in a pressure range of 0-15 GPa for all Bi2-xSbxTe3-ySey samples, and the maximum Tc was 5.45 K, recorded at 13.5 GPa in Bi2-xSbxTe3-ySey at x=0 and y=1.0. The magnetic field (H) dependence of the R-T plot for Bi2-xSbxTe3-ySey was measured to characterize the superconducting pairing mechanism of pressure-induced superconducting phase.