In this work, we develop a novel construction technique for set-theoretical solutions of the Yang-Baxter equation. Our technique, named the matched product, is an innovative tool to construct new classes of involutive solutions as the matched product of two involutive solutions is still involutive, and vice versa. This method produces new examples of idempotent solutions as the matched product of other idempotent ones. We translate the construction in the context of semi-braces, which are algebraic structures tightly linked with solutions that generalize the braces introduced by Rump. In addition, we show that the solution associated to the matched product of two semi-braces is indeed the matched product of the solutions associated to those two semi-braces.