The van der Waals heterostructures of two-dimensional (2D) atomic crystals constitute a new paradigm in nanoscience. Hybrid devices of graphene with insulating 2D hexagonal boron nitride (h-BN) have emerged as promising nanoelectronic architectures through demonstrations of ultrahigh electron mobilities and charge-based tunnel transistors. Here, we expand the functional horizon of such 2D materials demonstrating the quantum tunneling of spin polarized electrons through atomic planes of CVD grown h-BN. We report excellent tunneling behavior of h-BN layers together with tunnel spin injection and transport in graphene using ferromagnet/h-BN contacts. Employing h-BN tunnel contacts, we observe enhancements in both spin signal amplitude and lifetime by an order of magnitude. We demonstrate spin transport and precession over micrometer-scale distances with spin lifetime up to 0.46 nanosecond. Our results and complementary magnetoresistance calculations illustrate that CVD h-BN tunnel barrier provides a reliable, reproducible and alternative approach to address the conductivity mismatch problem for spin injection into graphene.