In recent years, deep learning models have been successfully employed for augmenting low-resolution cosmological simulations with small-scale information, a task known as "super-resolution". So far, these cosmological super-resolution models have relied on generative adversarial networks (GANs), which can achieve highly realistic results, but suffer from various shortcomings (e.g. low sample diversity). We introduce denoising diffusion models as a powerful generative model for super-resolving cosmic large-scale structure predictions (as a first proof-of-concept in two dimensions). To obtain accurate results down to small scales, we develop a new "filter-boosted" training approach that redistributes the importance of different scales in the pixel-wise training objective. We demonstrate that our model not only produces convincing super-resolution images and power spectra consistent at the percent level, but is also able to reproduce the diversity of small-scale features consistent with a given low-resolution simulation. This enables uncertainty quantification for the generated small-scale features, which is critical for the usefulness of such super-resolution models as a viable surrogate model for cosmic structure formation.