Charm meson oscillations are observed in a time-dependent analysis of the ratio of D0→K+π−π+π− to D0→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 rK3πD and the product of the coherence factor RK3πD and a charm mixing parameter y′K3π. The constraints measured are rK3πD=(5.67±0.12)×10−2, which is the most precise determination to date, and RK3πDy′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 D0→K+π−π+π− branching fraction, and the first observation of D0-¯D0 oscillations in this decay mode, with a significance of 8.2 standard deviations.