We show that in gapped bilayer graphene, quasiparticle tunneling and the corresponding Berry phase can be controlled such that they exhibit features of single-layer graphene such as Klein tunneling. The Berry phase is detected by a high-quality Fabry-Pérot interferometer based on bilayer graphene. By raising the Fermi energy of the charge carriers, we find that the Berry phase can be continuously tuned from 2π down to 0.68π in gapped bilayer graphene, in contrast to the constant Berry phase of 2π in pristine bilayer graphene. Particularly, we observe a Berry phase of π, the standard value for single-layer graphene. As the Berry phase decreases, the corresponding transmission probability of charge carriers at normal incidence clearly demonstrates a transition from anti-Klein tunneling to nearly perfect Klein tunneling.