Control charts are sophisticated graphical tools used to detect and control aberrant variations. Different control schemes are designed to continuously monitor and improve the process stability and performance. This study proposes a bivariate exponentially weighted moving average chart for joint monitoring of the mean vector of Gumbel’s bivariate geometric (GBG) data. The performance of the proposed chart is compared with the Hotelling’s T2 chart. The results of the study indicated that the proposed control chart performs uniformly and substantially better than the Hotelling’s T2 chart. In addition to two real-life examples, an example based on simulated data is also considered and compared to existing charts to verify the superiority of the proposed chart. Based on the comparisons, it turns out that the MEWMA (GBG) chart outperforms the Hotelling’s T2 chart and paired individual EWMA control chart.