As opposed to conventional training methods tailored to minimize a given statistical metric or task-agnostic loss (e.g., mean squared error), Decision-Focused Learning (DFL) trains machine learning models for optimal performance in downstream decision-making tools. We argue that DFL can be leveraged to learn the parameters of system dynamics, expressed as constraint of the convex optimization control policy, while the system control signal is being optimized, thus creating an end-to-end learning framework. This is particularly relevant for systems in which behavior changes once the control policy is applied, hence rendering historical data less applicable. The proposed approach can perform system identification - i.e., determine appropriate parameters for the system analytical model - and control simultaneously to ensure that the model's accuracy is focused on areas most relevant to control. Furthermore, because black-box systems are non-differentiable, we design a loss function that requires solely to measure the system response. We propose pre-training on historical data and constraint relaxation to stabilize the DFL and deal with potential infeasibilities in learning. We demonstrate the usefulness of the method on a building Heating, Ventilation, and Air Conditioning day-ahead management system for a realistic 15-zone building located in Denver, US. The results show that the conventional RC building model, with the parameters obtained from historical data using supervised learning, underestimates HVAC electrical power consumption. For our case study, the ex-post cost is on average six times higher than the expected one. Meanwhile, the same RC model with parameters obtained via DFL underestimates the ex-post cost only by 3%.