We report the detection of 21 cm emission at an average redshift $\bar{z} = 2.3$ in the cross-correlation of data from the Canadian Hydrogen Intensity Mapping Experiment (CHIME) with measurements of the Lyman-$\alpha$ forest from eBOSS. Data collected by CHIME over 88 days in the $400-500$~MHz frequency band ($1.8 < z < 2.5$) are formed into maps of the sky and high-pass delay filtered to suppress the foreground power, corresponding to removing cosmological scales with $k_\parallel \lesssim 0.13\ \text{Mpc}^{-1}$ at the average redshift. Line-of-sight spectra to the eBOSS background quasar locations are extracted from the CHIME maps and combined with the Lyman-$\alpha$ forest flux transmission spectra to estimate the 21 cm-Lyman-$\alpha$ cross-correlation function. Fitting a simulation-derived template function to this measurement results in a $9\sigma$ detection significance. The coherent accumulation of the signal through cross-correlation is sufficient to enable a detection despite excess variance from foreground residuals $\sim6-10$ times brighter than the expected thermal noise level in the correlation function. These results are the highest-redshift measurement of \tcm emission to date, and set the stage for future 21 cm intensity mapping analyses at $z>1.8$.