The B+→D+K+π- decay is observed in a data sample corresponding to 3.0 fb-1 of pp collision data recorded by the LHCb experiment during 2011 and 2012. The signal significance is 8σ and the branching fraction is measured to be B(B+→D+K+π-)=(5.31±0.90±0.48±0.35)×10-6, where the uncertainties are statistical, systematic and due to the normalization mode B+→D-K+π+, respectively. The Dalitz plot appears to be dominated by broad structures. Angular distributions are exploited to search for quasi-two-body contributions from B+→D2∗(2460)0K+ and B+→D+K∗(892)0 decays. No significant signals are observed and upper limits are set on their branching fractions.