The first observation of the singly Cabibbo-suppressed Ω_{c}^{0}→Ω^{-}K^{+} and Ω_{c}^{0}→Ξ^{-}π^{+} decays is reported, using proton-proton collision data at a center-of-mass energy of 13 TeV, corresponding to an integrated luminosity of 5.4 fb^{-1}, collected with the LHCb detector between 2016 and 2018. The branching fraction ratios are measured to be B(Ω_{c}^{0}→Ω^{-}K^{+})/B(Ω_{c}^{0}→Ω^{-}π^{+})=[6.08±0.51(stat)±0.40(syst)]%,B(Ω_{c}^{0}→Ξ^{-}π^{+})/B(Ω_{c}^{0}→Ω^{-}π^{+})=[15.81±0.87(stat)±0.44(syst)±0.16(ext)]%. In addition, using the Ω_{c}^{0}→Ω^{-}π^{+} decay channel, the Ω_{c}^{0} baryon mass is measured to be M(Ω_{c}^{0})=2695.28±0.07(stat)±0.27(syst)±0.30(ext) MeV, improving the precision of the previous world average by a factor of 4.