A serious limitation in the study of many globular clusters -- especially those located near the Galactic Center -- has been the existence of large and differential extinction by foreground dust. In a series of papers we intend to map the differential extinction and remove its effects, using a new dereddening technique, in a sample of clusters in the direction of the inner Galaxy, observed using the Magellan 6.5m telescope and the Hubble Space Telescope. These observations and their analysis will let us produce high quality color-magnitude diagrams of these poorly studied clusters that will allow us to determine these clusters' relative ages, distances and chemistry and to address important questions about the formation and the evolution of the inner Galaxy. We also intend to use the maps of the differential extinction to sample and characterize the interstellar medium along the numerous low latitude lines of sight where the clusters in our sample lie. In this first paper we describe in detail our dereddening method along with the powerful statistics tools that allow us to apply it, and we show the kind of results that we can expect, applying the method to M62, one of the clusters in our sample. The width of the main sequence and lower red giant branch narrows by a factor of 2 after applying our dereddening technique, which will significantly help to constrain the age, distance, and metallicity of the cluster. ; Comment: Accepted for publication in AJ