Electric-field modulation of magnetism in strain-mediated multiferroic heterostructures is considered a promising scheme for enabling memory and magnetic microwave devices with ultralow power consumption. However, it is not well understood how electric-field-induced strain influences magnetic relaxation, an important physical process for device applications. Here, we investigate resonant magnetization dynamics in ferromagnet/ferroelectric multiferroic heterostructures, FeGaB/PMN-PT and NiFe/PMN-PT, in two distinct strain states provided by electric-field-induced ferroelectric phase transition. The strain not only modifies magnetic anisotropy but also magnetic relaxation. In FeGaB/PMN-PT, we observe a nearly two-fold change in intrinsic Gilbert damping by electric field, which is attributed to strain-induced tuning of spin-orbit coupling. By contrast, a small but measurable change in extrinsic linewidth broadening is attributed to inhomogeneous ferroelastic domain switching during the phase transition of the PMN-PT substrate.