A high-quality junction between graphene and metallic contacts is crucial in the creation of high-performance graphene transistors. In an ideal metal-graphene junction, the contact resistance is determined solely by the number of conduction modes in graphene. However, as yet, measurements of contact resistance have been inconsistent, and the factors that determine the contact resistance remain unclear. Here, we report that the contact resistance in a palladium-graphene junction exhibits an anomalous temperature dependence, dropping significantly as temperature decreases to a value of just 110 ± 20 Ω µm at 6 K, which is two to three times the minimum achievable resistance. Using a combination of experiment and theory we show that this behaviour results from carrier transport in graphene under the palladium contact. At low temperature, the carrier mean free path exceeds the palladium-graphene coupling length, leading to nearly ballistic transport with a transfer efficiency of ~75%. As the temperature increases, this carrier transport becomes less ballistic, resulting in a considerable reduction in efficiency.