Charm meson oscillations are observed in a time-dependent analysis of the ratio of D^{0}→K^{+}π^{-}π^{+}π^{-} to D^{0}→K^{-}π^{+}π^{-}π^{+} decay rates, using data corresponding to an integrated luminosity of 3.0  fb^{-1} recorded by the LHCb experiment. The measurements presented are sensitive to the phase-space averaged ratio of doubly Cabibbo-suppressed to Cabibbo-favored amplitudes r_{D}^{K3π} and the product of the coherence factor R_{D}^{K3π} and a charm mixing parameter y_{K3π}^{'}. The constraints measured are r_{D}^{K3π}=(5.67±0.12)×10^{-2}, which is the most precise determination to date, and R_{D}^{K3π}y_{K3π}^{'}=(0.3±1.8)×10^{-3}, which provides useful input for determinations of the CP-violating phase γ in B^{±}→DK^{±}, D→K^{∓}π^{±}π^{∓}π^{±} decays. The analysis also gives the most precise measurement of the D^{0}→K^{+}π^{-}π^{+}π^{-} branching fraction, and the first observation of D^{0}-D[over ¯]^{0} oscillations in this decay mode, with a significance of 8.2 standard deviations.