The Cabibbo-suppressed decay $\Lambda_b^0\rightarrow\chi_{c1}p\pi^-$ is observed for the first time using data from proton-proton collisions corresponding to an integrated luminosity of 6fb$^{-1}$, collected with the LHCb detector at a centre-of-mass energy of 13TeV. Evidence for the $\Lambda_b^0\rightarrow\chi_{c2}p\pi^-$ decay is also found. Using the $\Lambda_b^0\rightarrow\chi_{c1}pK^-$ decay as normalisation channel, the ratios of branching fractions are measured to be $$\begin{array}{rcl} \frac{ \mathcal{B} (\Lambda_b^0\rightarrow\chi_{c1}p\pi^-)}{\mathcal{B} (\Lambda_b^0\rightarrow\chi_{c1}pK^-)} & = & (6.59 \pm 1.01 \pm 0.22 ) \times 10^{-2} \,, \frac{\mathcal{B} (\Lambda_b^0\rightarrow\chi_{c2}p\pi^-)}{\mathcal{B} (\Lambda_b^0\rightarrow\chi_{c1}p\pi^-)} & = & 0.95 \pm 0.30 \pm 0.04 \pm 0.04 \,, \frac{\mathcal{B} (\Lambda_b^0\rightarrow\chi_{c2}pK^-)}{\mathcal{B} (\Lambda_b^0\rightarrow\chi_{c1}pK^-)} & = & 1.06 \pm 0.05 \pm 0.04 \pm 0.04 \,,\end{array}$$ where the first uncertainty is statistical, the second is systematic and the third is due to the uncertainties in the branching fractions of $\chi_{c1,2}\rightarrow J/\psi\gamma$ decays.