We present VIGS-SLAM, a visual-inertial 3D Gaussian Splatting SLAM system that achieves robust real-time tracking and high-fidelity reconstruction. Although recent 3DGS-based SLAM methods achieve dense and photoreal-istic mapping, their purely visual design degrades under motion blur, low texture, and exposure variations. Our method tightly couples visual and inertial cues within a unified optimization framework, jointly refining camera poses, depths, and IMU states. It features robust IMU initialization, time-varying bias modeling, and loop closure with consistent Gaussian updates. Experiments on four challenging datasets demonstrate our superiority over state-of-the-art methods.