Content Delivery Networks (CDNs) are critical for providing good user experience of cloud services. CDN providers typically collect various multivariate Key Performance Indicators (KPIs) time series to monitor and diagnose system performance. State-of-the-art anomaly detection methods mostly use deep learning to extract the normal patterns of data, due to its superior performance. However, KPI data usually exhibit non-additive Gaussian noise, which makes it difficult for deep learning models to learn the normal patterns, resulting in degraded performance in anomaly detection. In this paper, we propose a robust and noise-resilient anomaly detection mechanism using multivariate KPIs. Our key insight is that different KPIs are constrained by certain time-invariant characteristics of the underlying system, and that explicitly modelling such invariance may help resist noise in the data. We thus propose a novel anomaly detection method called SDFVAE, short for Static and Dynamic Factorized VAE, that learns the representations of KPIs by explicitly factorizing the latent variables into dynamic and static parts. Extensive experiments using real-world data show that SDFVAE achieves a F1-score ranging from 0.92 to 0.99 on both regular and noisy dataset, outperforming state-of-the-art methods by a large margin.