We present the first measurement of the cross section of Cabibbo-suppressed $\Lambda$ baryon production, using data collected with the MicroBooNE detector when exposed to the neutrinos from the Main Injector beam at the Fermi National Accelerator Laboratory. The data analyzed correspond to $2.2 \times 10^{20}$ protons on target of neutrino mode running and $4.9 \times 10^{20}$ protons on target of anti-neutrino mode running. An automated selection is combined with hand scanning, with the former identifying five candidate $\Lambda$ production events when the signal was unblinded, consistent with the GENIE prediction of $5.3 \pm 1.1$ events. Several scanners were employed, selecting between three and five events, compared with a prediction from a blinded Monte Carlo simulation study of $3.7 \pm 1.0$ events. Restricting the phase space to only include $\Lambda$ baryons that decay above MicroBooNE's detection thresholds, we obtain a flux averaged cross section of $2.0^{+2.2}_{-1.7} \times 10^{-40}$ cm$^2/$Ar, where statistical and systematic uncertainties are combined.