We address the problem of time-frequency audio inpainting, where the goal is to fill missing spectrogram portions with reliable information. Despite recent advances, existing approaches still face limitations in both reconstruction quality and computational efficiency. To bridge this gap, we propose a method that utilizes a phase-aware signal prior which exploits estimates of the instantaneous frequency. An optimization problem is formulated and solved using the generalized Chambolle-Pock algorithm. The proposed method is evaluated against other time-frequency inpainting methods, specifically a deep-prior audio inpainting neural network and the autoregression-based approach known as Janssen-TF. Our proposed approach surpassed these methods by a large margin in the objective evaluation as well as in the conducted subjective listening test, improving the state of the art. In addition, the reconstructions are obtained with a substantially reduced computational cost compared to alternative methods.