Manipulating materials at the nanometer scale is challenging, particularly if alignment with nanoscale electrodes is desired. Here we describe a lithography-free, self-aligned nanotrench ablation (SANTA) technique to create nanoscale trenches in a polymer like poly(methyl) methacrylate (PMMA). The nanotrenches are self-aligned with carbon nanotube (CNT) and graphene ribbon electrodes through a simple Joule heating process. Using simulations and experiments we investigate how the Joule power, ambient temperature, PMMA thickness, and substrate properties can improve the spatial resolution of this technique. We achieve sub-20 nm nanotrenches for the first time, by lowering the ambient temperature and reducing the PMMA thickness. We also demonstrate a functioning nanoscale resistive memory (RRAM) bit self-aligned with a CNT control device, achieved through the SANTA approach. This technique provides an elegant and inexpensive method to probe nanoscale devices using self-aligned electrodes, without the use of conventional alignment or lithography steps.