Inspired by the capability of exponentially-weighted moving average (EWMA) charts to balance sensitivity and false alarm rates, we propose one for zero-truncated Poisson processes. We present a systematic design and analytic framework for implementation. Further, we add a fast initial response (FIR) feature which ideally increases sensitivity without compromising false alarm rates. The proposed charts (basic and with FIR feature) were evaluated based on both in-control average run length (ARL0) to measure false alarm rate and out-of-control average run length (ARL1) to measure sensitivity to detect unwanted shifts. The evaluation process used a Markov chain intensive simulation study at different settings for different weighting parameters (ω). Empirical results suggest that for both scenarios, the basic chart had: (1) exponentially increasing ARLs as a function of the chart threshold L; and (2) ARLs were longer for smaller ωs. Moreover, the added FIR feature has indeed improved ARL1 within the range of 5% - 55%, resulting to quicker shift detections at a relatively minimal loss in ARL0. These results were also compared to Shewhart and CUSUM control charts at similar settings, and it was observed that the EWMA charts generally performed better by striking a balance between higher ARL0 and lower ARL1. These advantages of the EWMA charts were more pronounced when larger shifts in the parameter λ happened. Finally, a case application in monitoring hospital surgical out-of-controls is presented to demonstrate its usability in a real-world setting.