The Cabibbo-suppressed decay $\Lambda^0_b\rightarrow\psi(2S)p\pi^-$ is observed for the first time using a data sample collected by the LHCb experiment in proton-proton collisions corresponding to 1.0, 2.0 and 1.9fb$^{-1}$ of integrated luminosity at centre-of-mass energies of 7, 8 and 13TeV, respectively. The $\psi(2S)$ mesons are reconstructed in the $\mu^+\mu^-$ final state. The~branching fraction with respect to that of the $\Lambda^0_b\rightarrow\psi(2S)pK^-$ decay mode is measured to be $$\frac{\mathcal{B}\left(\Lambda^0_b\rightarrow\psi(2S)p\pi^- \right)} {\mathcal{B}\left(\Lambda^0_b\rightarrow\psi(2S)pK^-\right)}=\left(11.4 \pm 1.3 \pm 0.2\right)\!\%\,,$$ where the first uncertainty is statistical and the second is systematic. The $\psi(2S)p$ and $\psi(2S)\pi^-$ mass spectra are investigated and no evidence for exotic resonances is found.