Automotive structural adhesives must show a steady toughness performance in the temperature range of −40 °C to 80 °C, considering their actual usage environments. Core-shell rubber (CSR) nanoparticles are known to enhance the toughness of epoxy systems. In this study, a CSR, pre-dispersed, diglycidyl epoxy of bisphenol A (DGEBA) mixture at 35 wt % (KDAD-7101, Kukdo Chemical, Seoul, Korea) was used as a toughener for an automotive structural epoxy adhesive system. A simple, single-component, epoxy system of DGEBA/dicyandiamide with a latent accelerator was adopted, where the CSR content of the system was controlled from 0 to 50 phr by the CSR mixture. To determine the curing conditions, we studied the curing behavior of the system by differential scanning calorimetry (DSC). Modulus variations of the cured bulk epoxies were studied using a dynamic mechanical analyzer (DMA) in the dual cantilever mode. The flexural modulus of the cured epoxies at various temperatures (−40, −10, 20, 50, and 80 °C) showed the same tendency as the DMA results, and as the flexural strength, except at 0 phr. On the other hand, the strain at break exhibited the opposite tendency to the flexural modulus. To study the adhesion behavior, we performed single-lap joint (SLJ) and impact wedge-peel (IWP) tests. As the CSR content increased, the strength of the SLJ and dynamic resistance to the cleavage of the IWP improved. In particular, the SLJ showed excellent strength at low temperatures (32.74 MPa at 50 phr @ −40 °C (i.e., an 190% improvement compared to 17.2 MPa at 0 phr @ −40 °C)), and the IWP showed excellent energy absorption at high temperatures (21.73 J at 50 phr @ 80 °C (i.e., a 976% improvement compared to 2.07 J at 0 phr @ 80 °C)). The results were discussed in relation to the changes in the properties of the bulk epoxy depending on the temperature and CSR content. The morphology of the fracture surface was also provided, which offered useful information for composition studies using the CSR/epoxy system.