Near-ternary eutectic Sn–Ag–Cu alloys are leading candidates for Pb-free solders. These alloys have three solid phases: β–Sn, Ag3Sn, and Cu6Sn5. Starting from the fully liquid state in solidifying near-eutectic Sn–Ag–Cu alloys, the equilibrium eutectic transformation is kinetically inhibited. The Ag3Sn phase nucleates with minimal undercooling, but the β–Sn phase requires a typical undercooling of 15 to 30 °C for nucleation. Because of this disparity in the required undercooling for nucleation, large, platelike Ag3Sn structures can grow rapidly within the liquid phase, before the final solidification of the solder joints. At lower cooling rates, the large Ag3Sn plates can subtend the entire cross section of solder joints and can significantly influence the mechanical deformation behavior of the solder joints under thermomechanical fatigue conditions. In this paper, it is demonstrated that the Ag3Sn plate formation can be inhibited, an important factor in assuring the reliability of solder joints composed of these alloys.