Cut-and-paste $K$-theory is a new variant of higher algebraic $K$-theory that has proven to be useful in problems involving decompositions of combinatorial and geometric objects, e.g., scissors congruence of polyhedra and reconstruction problems in graph theory. In this paper, we show that this novel machinery can also be used in the study of matroids. Specifically, via the $K$-theory of categories with covering families developed by Bohmann-Gerhardt-Malkiewich-Merling-Zakharevich, we realize the Tutte polynomial map of Brylawski (also known as the universal Tutte-Grothendieck invariant for matroids) as the $K_0$-homomorphism induced by a map of $K$-theory spectra.