We focus on developing efficient and reliable policy optimization strategies for robot learning with real-world data. In recent years, policy gradient methods have emerged as a promising paradigm for training control policies in simulation. However, these approaches often remain too data inefficient or unreliable to train on real robotic hardware. In this paper we introduce a novel policy gradient-based policy optimization framework which systematically leverages a (possibly highly simplified) first-principles model and enables learning precise control policies with limited amounts of real-world data. Our approach $1)$ uses the derivatives of the model to produce sample-efficient estimates of the policy gradient and $2)$ uses the model to design a low-level tracking controller, which is embedded in the policy class. Theoretical analysis provides insight into how the presence of this feedback controller addresses overcomes key limitations of stand-alone policy gradient methods, while hardware experiments with a small car and quadruped demonstrate that our approach can learn precise control strategies reliably and with only minutes of real-world data.