Standard video codecs are rate-distortion optimization machines, where distortion is typically quantified using PSNR versus the source. However, it is now widely accepted that increasing PSNR does not necessarily translate to better visual quality. In this paper, a better balance between perception and fidelity is targeted, in order to provide for significant rate savings over state-of-the-art standards-based video codecs. Specifically, pre- and postprocessing neural networks are proposed that enhance the coding efficiency of standard video codecs when benchmarked with an array of well-established perceptual quality scores. These neural wrapper'' elements are end-to-end trained with a neural codec module serving as a differentiable proxy for standard video codecs. The codec proxy is jointly optimized with the pre- and post components, via a novel two-phase pretraining strategy and end-to-end iterative refinement with stop-gradient. This allows the neural pre- and postprocessor to learn to embed, remove and recover information in a codec-aware manner, thus improving its rate-quality performance. A single neural-wrapper model is thereby established and used for the entire rate-quality curve without needing any downscaling or upscaling. The trained model is tested with the AV1 and VVC standard codecs via an array of well-established objective quality scores (SSIM, MS-SSIM, VMAF, AVQT), as well as mean opinion scores (MOS) derived from ITU-T P.910 subjective testing. Experimental results show that the proposed approach improves all quality scores, with -18.5\% average Bjontegaard Delta-rate (BD-rate) saving over all objective scores and MOS improvement over both standard codecs. This illustrates the significant potential of neural wrapper components over standards-based video coding.