Here we report the discovery of pressure-induced superconductivity in quasi-one-dimensional ${\mathrm{Ta}}_{2}{\mathrm{PdSe}}_{6}$, through a combination of electrical transport, synchrotron x-ray diffraction, and theoretical calculations. Our transport measurements show that the superconductivity appears at a critical pressure ${P}_{\mathrm{c}}\ensuremath{\sim}18.3\phantom{\rule{0.16em}{0ex}}\mathrm{GPa}$ and is robust upon further compression up to 62.6 GPa. The estimated upper critical field ${\ensuremath{\mu}}_{0}{H}_{\mathrm{c}2}(0)$ in the pressurized ${\mathrm{Ta}}_{2}{\mathrm{PdSe}}_{6}$ is much lower than the Pauli limiting field, in contrast to the case in its isostructural analogs ${M}_{2}{\mathrm{Pd}}_{\mathrm{x}}{X}_{5}$ $(M=\mathrm{Nb}, \mathrm{Ta}; X=\mathrm{S}, \mathrm{Se})$. Concomitant with the occurrence of superconductivity, anomalies in pressure-dependent transport properties are observed, including sign reversal of Hall coefficient, abnormally enhanced resistance, and dramatically suppressed magnetoresistance. Meanwhile, room-temperature synchrotron x-ray diffraction experiments reveal the stability of the pristine monoclinic structure (space group $C2$/$m$) upon compression. Combined with the density functional theory calculations, we argue that a pressure-induced Lifshitz transition could be the electronic origin of the emergent superconductivity in ${\mathrm{Ta}}_{2}{\mathrm{PdSe}}_{6}$.