Sleep staging is pivotal for assessing sleep quality and clinical diagnostics. While contactless radio-frequency (RF) sensing offers an unobtrusive alternative to Polysomnography (PSG), existing methods often struggle with generalization across diverse devices and environments due to the scarcity of annotated RF data. To overcome this limitation, we propose XSensorSleep, a cross-sensor domain generalization framework. Uniquely, XSensorSleep is trained exclusively on large-scale respiratory datasets from chest and abdominal belts and is directly transferable to RF-derived respiratory signals. To bridge the domain gap, we partition the training data into multiple pseudo-domains based on data sources and sensor types, forcing the model to extract domain-invariant features from heterogeneous respiratory patterns. Furthermore, we introduce a hierarchical alignment strategy: Epoch-Level Feature Alignment (ELFA) suppresses sensor-specific morphological artifacts, while Spectral Temporal Alignment (STA) captures invariant global sleep architectures. Specifically, STA leverages the rotation-invariance of eigenvalue spectra to align whole-night transition dynamics, ensuring the model prioritizes universal physiological rhythms over device-dependent signal fluctuations. Extensive evaluations across various home and clinical RF datasets demonstrate that XSensorSleep achieves superior generalization, enabling practical, automated, and label-free non-contact sleep staging.