Magnetic tunnel junction (MTJ) is a leading contender for next generation high-density nonvolatile memory technology. Fast and efficient switching of MTJs between different resistance states is a challenging problem, which can be tackled by using an unconventional stimulus-a femtosecond laser pulse. Herein, we report an experimental study of the laser-induced magnetization dynamics in a Co20Fe60B20/MgO/Co20Fe60B20 (CoFeB/MgO/CoFeB) MTJ with ultrathin CoFeB electrodes possessing perpendicular magnetic anisotropy (PMA). In addition to ultrafast demagnetization, a femtosecond laser pulse gives rise to a decaying magnetization precession in the thinner CoFeB layer subjected to an in-plane magnetic field, while the magnetization of the thicker CoFeB layer remains aligned with the applied field. Remarkably, the precession frequency demonstrates a strong and nonlinear rise with increasing pump fluence, which stems from the complete laser-induced suppression of PMA in the 1.2 nm-thick CoFeB electrode reached at a moderate fluence of about 1.8 mJ cm-2 at room temperature. This important feature signifies that the laser excitation of such an electrode can enable an ultrafast transition from a perpendicular-to-plane to an in-plane magnetization orientation in the absence of a magnetic field and reveals the feasibility of the laser-driven switching of MTJ between different states. The revealed gradual quenching of PMA with increasing fluence is explained by the laser-induced heating of the MTJ, which affects the interfacial magnetic anisotropy stronger than the shape anisotropy. Interestingly, at low fluences, the values of interfacial anisotropy and saturation magnetization altered by the laser excitation scale with each other as expected for the two-site anisotropic exchange interaction, but the scaling exponent increases significantly at moderate fluences, which enables the realization of a laser-induced spin reorientation transition.