The reliability of modern applications increasingly depends on the stability of underlying distributed web services. Monitoring these services through time series analysis is essential for detecting performance degradations and security anomalies. However, many existing anomaly detection models struggle to adapt to the dynamic nature of cloud-based environments, particularly when dealing with evolving service patterns or system reconfigurations. To help address these challenges, this paper proposes DeLSTM-AE, a novel framework that integrates time series decomposition with a Long Short-Term Memory (LSTM) and an Autoencoder (AE) to improve the detection of anomalies in univariate time series data. By separating seasonal, trend, and residual components prior to training, our model enables more targeted learning of distinct temporal structures, thereby improving its ability to capture subtle anomalies. We evaluate the proposed framework on the benchmark AIOps Challenge 2018 dataset, which includes real-time Key Performance Indicator (KPI) logs from major cloud service providers such as eBay, Sogou, and Tencent. DeLSTM-AE achieves superior performance compared to existing state-of-the-art methods, with a Precision of 99%, Recall of 94%, F1 score of 96%, and Accuracy of 94% across multiple KPI test sets. The results demonstrate that decomposition-based preprocessing substantially enhances both anomaly detection accuracy and model generalisation, offering a scalable and effective solution for real-world monitoring systems.