In this study, thin ribbons of amorphous Mg72Zn27Pt1 and Mg72Zn27Ag1 alloys with potential use in biomedicine were analyzed in terms of the crystallization mechanism. Non-isothermal annealing in differential scanning calorimetry (DSC) with five heating rates and X-ray diffraction (XRD) during heating were performed. Characteristic temperatures were determined, and the relative crystalline volume fraction was estimated. The activation energies were calculated using the Kissinger method and the Avrami exponent using the Jeziorny–Avrami model. The addition of platinum and silver shifts the onset of crystallization towards higher temperatures, but Pt has a greater impact. In each case, Eg > Ex > Ep (activation energy of the glass transition, the onset of crystallization, and the peak, respectively), which indicates a greater energy barrier during glass transition than crystallization. The highest activation energy was observed for Mg72Zn27Pt1 due to the difference in the size of the atoms of all alloy components. The crystallization in Mg72Zn27Ag1 occurs faster than in Mg72Zn27Pt1, and the alloy with Pt has higher (temporary) thermal stability. The Avrami exponent (n) values oscillate in the range of 1.7–2.6, which can be interpreted as one- and two-dimensional crystal growth with a constant/decreasing nucleation rate during the process. Moreover, the lower the heating rate, the higher the nucleation rate. The values of n for Mg72Zn27Pt1 indicate a greater number of nuclei and grains than for Mg72Zn27Ag1. The XRD tests indicate the presence of α-Mg and Mg12Zn13 for both Mg72Zn27Pt1 and Mg72Zn27Ag1, but the contribution of the Mg12Zn13 phase is greater for Mg72Zn27Ag1