We report on a measurement of the flavor-specific \(B^{0}_{s}\) lifetime and of the \(D^{-}_{s}\) lifetime using proton-proton collisions at center-of-mass energies of 7 and 8 TeV, collected by the LHCb experiment and corresponding to 3.0 fb\(^{-1}\) of integrated luminosity. Approximately 407 000 \(B^{0}_{s} \rightarrow D^{(*)-}_{s} \mu^{+}\nu_\mu \) decays are partially reconstructed in the \(K^{+}K^{-}\pi^{-}\mu^{+}\) final state. The \(B^{0}_{s}\) and \(D^{-}_{s}\) natural widths are determined using, as a reference, kinematically similar \(B^{0} \rightarrow D^{(*)-}\mu^{+}\nu_\mu\) decays reconstructed in the same final state. The resulting differences between widths of \(B^{0}_{s}\) and \(B^{0}\) mesons and of \(D^{-}_{s}\) and \(D^{-}\) mesons are \(\Delta_\Gamma(B) =-0.0115 \pm 0.0053 (stat) \pm 0.0041 (syst)\) ps\(^{-1}\) and \(\Delta_\Gamma(D) = 1.0131 \pm 0.0117 (stat) \pm 0.0065 (syst)\) ps\(^{-1}\), respectively. Combined with the known \(B^{0}\) and \(D^{-}\) lifetimes, these yield the flavor-specific \(B^{0}_{s}\) lifetime, \(\tau^{\rm fs}_{B^{0}_{s}} = 1.547 \pm 0.013 (stat) \pm 0.010 (syst) \pm 0.004 (\tau_{B})\) ps and the \(D^{-}_{s}\) lifetime, \(\tau_{D^{-}_{s}} = 0.5064 \pm 0.0030 (stat) \pm 0.0017 (syst) \pm 0.0017 (\tau_{D})\) ps The last uncertainties originate from the limited knowledge of the \(B^0\) and \(D^{-}\) lifetimes. The results improve upon current determinations.