With the development and popularity of sensors installed in manufacturing systems, complex data are collected during manufacturing processes, which brings challenges for traditional process control methods. This paper proposes a novel process control and monitoring method for the complex structure of high-dimensional image-based overlay errors (modeled in tensor form), which are collected in semiconductor manufacturing processes. The proposed method aims to reduce overlay errors using limited control recipes. We first build a high-dimensional process model and propose different tensor-on-vector regression algorithms to estimate parameters in the model to alleviate the curse of dimensionality. Then, based on the estimate of tensor parameters, the exponentially weighted moving average (EWMA) controller for tensor data is designed whose stability is theoretically guaranteed. Considering the fact that low-dimensional control recipes cannot compensate for all high-dimensional disturbances on the image, control residuals are monitored to prevent significant drifts of uncontrollable high-dimensional disturbances. Through extensive simulations and real case studies, the performances of parameter estimation algorithms and the EWMA controller in tensor space are evaluated. Compared with existing image-based feedback controllers, the superiority of our method is verified especially when disturbances are not stable.