The B+→D∗−K+π+ decay potentially provides an excellent way to investigate charm meson spectroscopy. The decay is searched for in a sample of proton-proton collision data collected with the LHCb detector at center-of-mass energies of 7 and 8 TeV, corresponding to an integrated luminosity of 3 fb−1. A clear signal is observed, and the ratio of its branching fraction to that of theB+→D∗−K+π+ normalization channel is measured to be B(B+→D∗−K+π+)/B(B+→D∗−π+π+)=(6.39±0.27±0.48)×10−2, where the first uncertainty is statistical and the second is systematic. This is the first observation of the B+→D∗−K+π+ decay