Graphene-like materials provide a unique opportunity to explore quantum-relativistic phenomena in a condensed matter laboratory. Interesting phenomena associated with the internal degrees of freedom, spin and valley, including quantum spin-Hall effect, have been theoretically proposed, but could not be observed so far largely due to disorder and density inhonogeneity. We show that weak magnetic field breaks the symmetries that protect flavor (spin, valley) degeneracy, and induces large bulk non-quantized flavor-Hall effect in graphene. The effect occurs due to flavor splitting which generates the imbalance of the Hall resistivities of the two flavor species. At the Dirac point, flavor-Hall effect is greatly enhanced due to anomalous magnetotransport of Dirac-like carriers. The flavor-Hall effect is robust in the presence of disorder and interactions, and can serve as a hallmark of the quasi-relativistic carriers in the system. It manifests itself in large nonlocal transport mediated by long-lived flavor currents, as well as in flavor injection and accumulation experiments. Recent experimental observation of the giant nonlocal transport in graphene following our prediction opens up new opportunities for creating and manipulating flavor currents. Our work also shows that nonlocal electric transport provides a tool to study neutral modes in solid-states systems.