A noise-suppressing low-light image enhancement approach is proposed in this paper based on the extent of exposedness at each image pixel. To this end, a progressive, structure-aware exposedness estimation procedure is presented that quantifies local and global exposedness. These exposedness values are leveraged to produce a locally smooth pixel-level map that signifies the required degrees of enhancement at image pixels. This map is subsequently used in an enhancement function, which satisfies a few important properties, to generate the enhanced image. Before the enhancement, inherent noise in the low-light image is diminished employing a detail-preserving, low gradient magnitude suppression method. Subjective and quantitative analysis of results on a wide variety of natural and synthetically generated low-light images from standard databases using PSNR, iRSE, SSIM, and measures of perceptual quality, natural image statistics and brightness preservation suggests that our approach in general outperforms the state-of-the-art. Ablation studies and further experiments show the importance of a few components of our approach, and that our approach is computationally fast.