Experimental engineering of high-dimensional quantum states is a crucial task for several quantum information protocols. However, a high degree of precision in the characterization of the noisy experimental apparatus is required to apply existing quantum-state engineering protocols. This is often lacking in practical scenarios, affecting the quality of the engineered states. We implement, experimentally, an automated adaptive optimization protocol to engineer photonic orbital angular momentum (OAM) states. The protocol, given a target output state, performs an online estimation of the quality of the currently produced states, relying on output measurement statistics, and determines how to tune the experimental parameters to optimize the state generation. To achieve this, the algorithm does not need to be imbued with a description of the generation apparatus itself. Rather, it operates in a fully black-box scenario, making the scheme applicable in a wide variety of circumstances. The handles controlled by the algorithm are the rotation angles of a series of waveplates and can be used to probabilistically generate arbitrary four-dimensional OAM states. We showcase our scheme on different target states both in classical and quantum regimes and prove its robustness to external perturbations on the control parameters. This approach represents a powerful tool for automated optimizations of noisy experimental tasks for quantum information protocols and technologies.