The deployment of robot controllers is hindered by modeling discrepancies due to necessary simplifications for computational tractability or inaccuracies in data-generating simulators. Such discrepancies typically require ad-hoc tuning to meet the desired performance, thereby ensuring successful transfer to a target domain. We propose a framework for automated, gradient-based tuning to enhance performance in the deployment domain by leveraging differentiable simulators. Our method collects rollouts in an iterative manner to co-tune the simulator and controller parameters, enabling systematic transfer within a few trials in the deployment domain. Specifically, we formulate multi-step objectives for tuning and employ alternating optimization to effectively adapt the controller to the deployment domain. The scalability of our framework is demonstrated by co-tuning model-based and learning-based controllers of arbitrary complexity for tasks ranging from low-dimensional cart-pole stabilization to high-dimensional quadruped and biped tracking, showing performance improvements across different deployment domains.