Voltage/electric-field control of ultrafast magnetization dynamics in magnetoelectrics provides a novel avenue for electronic tuning with orders of magnitude less power consumption, improved tuning response time, and more compact form factor, compared with conventional magnetic field control of magnetization dynamics. Magnetoelectrically tuned laser-driven magnetization dynamics has the potential to enable the next generation of optomagnetic devices from THz communication to optical magnetic recording. In this study, we fabricated a magnetoelectric heterostructure, specifically ferromagnetic ${({\mathrm{Fe}}_{81}{\mathrm{Ga}}_{19})}_{88}{\mathrm{B}}_{12}$ (FeGaB) thin film deposited on a $\mathrm{Pb}({\mathrm{Mg}}_{1/3}\phantom{\rule{0.16em}{0ex}}{\mathrm{Nb}}_{2/3}){\mathrm{O}}_{3}\text{\ensuremath{-}}\mathrm{PbTi}{\mathrm{O}}_{3}$ ferroelectric substrate, to explore the electric field tuning of ultrafast demagnetization and to understand how magnetic anisotropy changes postdemagnetization. We characterized the magnetoelectric coupling of our heterostructure to demonstrate the static and dynamic magnetization tunability with an applied electric field. We utilized the time-resolved magneto-optic Kerr effect to understand the ultrafast demagnetization process under different applied electric fields and magnetic fields. The typically observed strain-induced magnetic easy-axis rotation in a ferromagnetic/ferroelectric heterostructure was also observed to tune the ultrafast demagnetization of FeGaB in our experiment. Additionally, we found that the magnetization rotation can be achieved with a lower electric field compared with static tuning without laser heating. Furthermore, we observed the hysteresis loops postultrafast demagnetization and found that the magnetoelectric heterostructure exhibits a mixture of volatile and nonvolatile behaviors. These findings shed light on the potential of our magnetoelectric heterostructure for ultrafast optomagnetic devices and electric field tuning of spintronic THz emitters.