The first observation of the rare decay ${B}_{s}^{0}\ensuremath{\rightarrow}\ensuremath{\phi}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ and evidence for ${B}^{0}\ensuremath{\rightarrow}\ensuremath{\phi}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ are reported, using $pp$ collision data recorded by the LHCb detector at center-of-mass energies $\sqrt{s}=7$ and 8 TeV, corresponding to an integrated luminosity of $3\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$. The branching fractions in the ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ invariant mass range $400<m({\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}})<1600\text{ }\text{ }\mathrm{MeV}/{c}^{2}$ are $[3.48\ifmmode\pm\else\textpm\fi{}0.23\ifmmode\pm\else\textpm\fi{}0.17\ifmmode\pm\else\textpm\fi{}0.35]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ and $[1.82\ifmmode\pm\else\textpm\fi{}0.25\ifmmode\pm\else\textpm\fi{}0.41\ifmmode\pm\else\textpm\fi{}0.14]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}$ for ${B}_{s}^{0}\ensuremath{\rightarrow}\ensuremath{\phi}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ and ${B}^{0}\ensuremath{\rightarrow}\ensuremath{\phi}{\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ respectively, where the uncertainties are statistical, systematic and from the normalization mode ${B}_{s}^{0}\ensuremath{\rightarrow}\ensuremath{\phi}\ensuremath{\phi}$. A combined analysis of the ${\ensuremath{\pi}}^{+}{\ensuremath{\pi}}^{\ensuremath{-}}$ mass spectrum and the decay angles of the final-state particles identifies the exclusive decays ${B}_{s}^{0}\ensuremath{\rightarrow}\ensuremath{\phi}{f}_{0}(980)$, ${B}_{s}^{0}\ensuremath{\rightarrow}\ensuremath{\phi}{f}_{2}(1270)$, and ${B}_{s}^{0}\ensuremath{\rightarrow}\ensuremath{\phi}{\ensuremath{\rho}}^{0}$ with branching fractions of $[1.12\ifmmode\pm\else\textpm\fi{}0.1{6}_{\ensuremath{-}0.08}^{+0.09}\ifmmode\pm\else\textpm\fi{}0.11]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$, $[0.61\ifmmode\pm\else\textpm\fi{}0.1{3}_{\ensuremath{-}0.05}^{+0.12}\ifmmode\pm\else\textpm\fi{}0.06]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ and $[2.7\ifmmode\pm\else\textpm\fi{}0.7\ifmmode\pm\else\textpm\fi{}0.2\ifmmode\pm\else\textpm\fi{}0.2]\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}7}$, respectively.