Real-time anomaly detection in data streams requires continuous adaptation to evolving patterns and concept drift, yet existing methods rely on static algorithm selection and fixed hyperparameters that become suboptimal as data characteristics change. We introduce AutoSAD, the first fully autonomous framework that solves unsupervised streaming anomaly detection through intelligent model selection. Our approach maintains an ensemble of diverse detectors and employs multi-armed bandit optimization with normalized anomaly scores as reward signals, coupled with evolutionary hyperparameter mutation guided by performance feedback. Comprehensive evaluation on diverse datasets demonstrates that AutoSAD achieves superior performance, outperforming state-of-the-art streaming detectors and showing statistically significant improvements across varying data stream characteristics.