It is anticipated that future nanoelectronic devices will utilize carbon nanotubes (CNT) and/or single graphene sheets (SGS) as the low-level on-chip interconnects or functional elements. Here we address the contact resistance of Cu for higher level on-chip interconnects with CNT or SGS elements. We use first-principles quantum mechanical (QM) density functional and matrix Green's function methods to show that perfect Cu-SGS contact has a contact resistance of 16.3 MΩ for a one square nanometer contact. Then we analyzed possible improvements in contact resistance through incorporation of simple functional groups such as aryl (-C 6 H 4 -), acetylene (-CC-), carboxyl (-COO-), and amide (-CONH-), on CNT. We find that all four anchors enhance the interfacial mechanical stabilities and electrical conductivity. The best scenario is -COOH functionalized CNT which reduces the contact resistance to the Cu by a factor of 275 and increases the mechanical stability by 26 times.