The ${B}^{+}\ensuremath{\rightarrow}{{D}^{*}}^{\ensuremath{-}}{K}^{+}{\ensuremath{\pi}}^{+}$ 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 $3\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. A clear signal is observed, and the ratio of its branching fraction to that of the ${B}^{+}\ensuremath{\rightarrow}{D}^{*\ensuremath{-}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{+}$ normalization channel is measured to be $\mathcal{B}({B}^{+}\ensuremath{\rightarrow}{D}^{*\ensuremath{-}}{K}^{+}{\ensuremath{\pi}}^{+})/\mathcal{B}({B}^{+}\ensuremath{\rightarrow}{D}^{*\ensuremath{-}}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{+})=(6.39\ifmmode\pm\else\textpm\fi{}0.27\ifmmode\pm\else\textpm\fi{}0.48)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}2},$ where the first uncertainty is statistical and the second is systematic. This is the first observation of the ${B}^{+}\ensuremath{\rightarrow}{{D}^{*}}^{\ensuremath{-}}{K}^{+}{\ensuremath{\pi}}^{+}$ decay.