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 the B+→D∗-π+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.