Reverberation mapping is a robust method to measure the masses of supermassive black holes (SMBHs) outside of the local Universe. Measurements of the radius -- luminosity ($R-L$) relation using the Mg II emission line are critical for determining these masses near the peak of quasar activity at $z \approx 1 - 2$, and for calibrating secondary mass estimators based on Mg II that can be applied to large samples with only single-epoch spectroscopy. We present the first nine Mg II lags from our five-year Australian Dark Energy Survey (OzDES) reverberation mapping program, which substantially improves the number and quality of Mg II lag measurements. As the Mg II feature is somewhat blended with iron emission, we model and subtract both the continuum and iron contamination from the multi-epoch spectra before analyzing the Mg II line. We also develop a new method of quantifying correlated spectroscopic calibration errors based on our numerous, contemporaneous observations of F-stars. The lag measurements for seven of our nine sources are consistent with both the H$��$ and Mg II $R-L$ relations reported by previous studies. Our simulations verify the lag reliability of our nine measurements, and we estimate that the median false positive rate of the lag measurements is $4\%$.

18 pages, 12 figures. MNRAS, Volume 507, Issue 3, November 2021, Pages 3771-3788