We theoretically study the scattering of electrons (massive Dirac fermions) at a laternal potential step in bilayer graphene. Under certain conditions, the scattering shows perfect transmission or perfect reflection, which corresponds to the Klein tunneling in monolayer graphene. We attribute the perfect transmission/reflection to Berry phase pi (rather than the conventional value 2 pi of bilayer graphene) associated with the pseudospin-1/2 of the Dirac fermions. We discuss how to detect this Berry-phase effect in experiments.