The incorporation of advanced sensors and machine learning techniques has enabled modern manufacturing enterprises to perform data-driven classification-based anomaly detection based on the sensor data collected in manufacturing processes. However, one critical challenge is that newly presented defect category may manifest as the manufacturing process continues, resulting in monitoring performance deterioration of previously trained machine learning models. Hence, there is an increasing need for empowering machine learning models to learn continually. Among all continual learning methods, memory-based continual learning has the best performance but faces the constraints of data storage capacity. To address this issue, this paper develops a novel pseudo replay-based continual learning framework by integrating class incremental learning and oversampling-based data generation. Without storing all the data, the developed framework could generate high-quality data representing previous classes to train machine learning model incrementally when new category anomaly occurs. In addition, it could even enhance the monitoring performance since it also effectively improves the data quality. The effectiveness of the proposed framework is validated in three cases studies, which leverages supervised classification problem for anomaly detection. The experimental results show that the developed method is very promising in detecting novel anomaly while maintaining a good performance on the previous task and brings up more flexibility in model architecture.