High-resolution (HR) synthetic aperture radar (SAR) images play an important role in people’s daily life and military applications. However, due to the interference of speckle noise, the texture details of the SAR images become relatively blurred. The fine texture details can be reconstructed by increasing the resolution of the SAR images. Generative adversarial networks achieve high performance in image super-resolution (SR) reconstruction, but the existing generative adversarial networks only pay attention to the discrimination of HR images without that of the low-resolution (LR) images. If the reconstructed HR image is sufficiently realistic, the LR image obtained from downsampled super-resolved images should also be the same as the original LR image. To take advantage of the LR image, an SAR image SR reconstruction algorithm based on cross-resolution discrimination (CRD) using teacher–student network is proposed. First, the teacher discriminator network (TD-Net) discriminates the HR images, which enriches the reconstructed HR images with more high-frequency texture details. Second, the student discriminator network (SD-Net) discriminates the LR images, which enables the reconstructed HR images to be accurately downsampled to the original LR image. Finally, the TD-Net guides the training of the SD-Net by transmitting distillation knowledge to the SD-Net, which further improves the discriminative performance of the SD-Net. Experiments on the SAR image dataset demonstrate that the performance of the proposed CRD algorithm is better than other algorithms when both the objective evaluations and subjective effects are considered.