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-π0 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-π0 is measured to be (1.13±0.08stat±0.03syst)×10-3. The ratio of branching fractions of D+→K+π+π-π0 over D+→K-π+π+π0 is found to be (1.81±0.15)%, which corresponds to (6.28±0.52)tan4θC, where θC 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±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+2.5stat±0.2syst)×10-5.