Exploiting 2D materials for spintronic applications can potentially realize next-generation devices featuring low power consumption and quantum operation capability. The magnetic exchange field (MEF) induced by an adjacent magnetic insulator enables efficient control of local spin generation and spin modulation in 2D devices without compromising the delicate material structures. Using graphene as a prototypical 2D system, we demonstrate that its coupling to the model magnetic insulator (EuS) produces a substantial MEF (>14 T) with the potential to reach hundreds of tesla, which leads to orders-of-magnitude enhancement of the spin signal originating from the Zeeman spin Hall effect. Furthermore, the new ferromagnetic ground state of Dirac electrons resulting from the strong MEF may give rise to quantized spin-polarized edge transport. The MEF effect shown in our graphene/EuS devices therefore provides a key functionality for future spin logic and memory devices based on emerging 2D materials in classical and quantum information processing.