Using 2.93 fb^{-1} of e^{+}e^{-} collision data collected at a center-of-mass energy of 3.773 GeV with the BESIII detector, the first observation of the doubly Cabibbo-suppressed decay D^{+}→K^{+}π^{+}π^{-}π^{0} is reported. After removing decays that contain narrow intermediate resonances, including D^{+}→K^{+}η, D^{+}→K^{+}ω, and D^{+}→K^{+}ϕ, the branching fraction of the decay D^{+}→K^{+}π^{+}π^{-}π^{0} is measured to be (1.13±0.08_{stat}±0.03_{syst})×10^{-3}. The ratio of branching fractions of D^{+}→K^{+}π^{+}π^{-}π^{0} over D^{+}→K^{-}π^{+}π^{+}π^{0} is found to be (1.81±0.15)%, which corresponds to (6.28±0.52)tan^{4}θ_{C}, where θ_{C} is the Cabibbo mixing angle. This ratio is significantly larger than the corresponding ratios for other doubly Cabibbo-suppressed decays. The asymmetry of the branching fractions of charge-conjugated decays D^{±}→K^{±}π^{±}π^{∓}π^{0} is also determined, and no evidence for CP violation is found. In addition, the first evidence for the D^{+}→K^{+}ω decay, with a statistical significance of 3.3σ, is presented and the branching fraction is measured to be B(D^{+}→K^{+}ω)=(5.7_{-2.1}^{+2.5}_{stat}±0.2_{syst})×10^{-5}.