End-to-end autonomous driving has achieved remarkable progress by planning trajectories directly from raw sensor data. Yet the step from planning to control still relies on hand-tuned tracking controllers that block the gradient path from vehicle motion back to the network. This gap stems from non-differentiable vehicle dynamics, aliased supervision from coarsely sampled expert labels, rate mismatch between high-rate control and low-rate re-planning, and vehicle model discrepancies between training and deployment. This paper proposes DIVA (Differentiable Identified Vehicle dynamics for end-to-end Autonomous driving) to close this gap. An identified actuator-level vehicle model is embedded in the training loop of the end-to-end driving network, forming a differentiable chain from perception to vehicle states. A learning-based control decoder replaces the external controller and is supervised on the rolled-out vehicle states, bypassing the aliased control-input labels. A rate-consistent control-input parameterization reconciles the heterogeneous sampling rates and enables asynchronous execution with a 3.8 ms control-only cycle. On-policy re-identification and an online disturbance observer address training-time distribution shift and deployment-time model mismatch respectively. On the closed-loop Bench2Drive benchmark, DIVA achieves a driving score of 86.7, surpassing the widely adopted state-of-the-art approach by 3.3 points while nearly halving mean jerk and reducing brake switching by 64%, and maintains superior path tracking accuracy under significant vehicle model mismatch at deployment.