We measure the magnetotransport properties of the three-dimensional Dirac semimetal Cd_{3}As_{2} single crystal under magnetic fields up to 36 T. Shubnikov-de Haas (SdH) oscillations are clearly resolved and the n=1 Landau level is reached. A detailed analysis on the intercept of the Landau index plot reveals a significant dependence of the SdH phase factor on the orientation of the applied magnetic field. When the magnetic field is applied in the [001] direction, i.e., along the fourfold screw axis of the tetragonal crystal structure, a nontrivial π Berry phase, as predicted for the Dirac fermions, is observed. However, in a magnetic field tilted away from the [001] direction, the π Berry phase is evidently reduced, and a considerable enhancement of the effective mass is also revealed. Our observations demonstrate that the Dirac dispersion in Cd_{3}As_{2} is effectively modified in a tilted magnetic field, whereas the preserved π Berry phase in a magnetic field along the [001] direction can be related to the realization of the Weyl fermions. The sudden change of the SdH phase also indicates a possible topological phase transition induced by the symmetry-breaking effect.