We present a method to estimate and map the two-dimensional distribution of dust extinction in the late-type spiral galaxy NGC 959 from the theoretical and observed flux ratio of optical V and mid-IR (MIR) 3.6 μm images. Our method is applicable to both young and old stellar populations for a range of metallicities, and is not restricted to lines of sight toward star-formation (SF) regions. We explore this method using a pixel-based analysis on images of NGC 959 obtained in the V band at the Vatican Advanced Technology Telescope and at 3.6 μm (L band) with Spitzer/Infrared Array Camera. We present the original and extinction corrected Galaxy Evolution Explorer (GALEX) far-UV (FUV) and near-UV (NUV) images, as well as optical UBVR images of NGC 959. While the dust lanes are not clearly evident at GALEX resolution, our dust map clearly traces the dust that can be seen silhouetted against the galaxy's disk in the high-resolution Hubble Space Telescope (HST) images of NGC 959. The advantages of our method are (1) it only depends on two relatively common broadband images in the optical V band and in the MIR at 3.6 μm (but adding a near-UV band improves its fidelity); and (2) it is able to map the two-dimensional spatial distribution of dust within a galaxy. This powerful tool could be used to measure the detailed distribution of dust extinction within higher redshift galaxies to be observed with, e.g., the Hubble Space Telescope (HST)/WFC3 (optical near-IR) and James Webb Space Telescope (mid-IR), and to distinguish properties of dust within galaxy bulges, spiral arms, and inter-arm regions.