The first observation of the decay ${B}^{0}\ensuremath{\rightarrow}{D}^{0}{\overline{D}}^{0}{K}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ is reported using proton-proton collision data corresponding to an integrated luminosity of $4.7\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ collected by the LHCb experiment in 2011, 2012 and 2016. The measurement is performed in the full kinematically allowed range of the decay outside of the ${D}^{*\ensuremath{-}}$ region. The ratio of the branching fraction relative to that of the control channel ${B}^{0}\ensuremath{\rightarrow}{D}^{*\ensuremath{-}}{D}^{0}{K}^{+}$ is measured to be $\mathcal{R}=(14.2\ifmmode\pm\else\textpm\fi{}1.1\ifmmode\pm\else\textpm\fi{}1.0)%$, where the first uncertainty is statistical and the second is systematic. The absolute branching fraction of ${B}^{0}\ensuremath{\rightarrow}{D}^{0}{\overline{D}}^{0}{K}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ decays is thus determined to be $\mathcal{B}({B}^{0}\ensuremath{\rightarrow}{D}^{0}{\overline{D}}^{0}{K}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})=(3.50\ifmmode\pm\else\textpm\fi{}0.27\ifmmode\pm\else\textpm\fi{}0.26\ifmmode\pm\else\textpm\fi{}0.30)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4}$, where the third uncertainty is due to the branching fraction of the control channel. This decay mode is expected to provide insights to spectroscopy and the charm-loop contributions in rare semileptonic decays.