We study phase-controlled quantum entanglement and one-way quantum steering in a cavity magnonic system, where two magnon modes couple to two microwave cavities driven by squeezed fields. The magnon-magnon entanglement and the steering directivity can be controlled by adjusting the ratio of the coupling strengths between cavity modes and magnon modes. In particular, the quantum entanglement and one-way quantum steering can be significantly enhanced by the squeezed vacuum field, which also exhibits robustness against temperature variations. More intriguingly, both quantum entanglement and steering between two magnon modes are susceptible to variations in the phase difference of two Josephson parametric amplifiers, and thus the phase-controlled quantum entanglement and one-way quantum steering can be achieved. Our scheme may have potential applications in quantum information.