A recent breakthrough by Ambainis, Balodis, Iraids, Kokainis, Prūsis and Vihrovs (SODA'19) showed how to construct faster quantum algorithms for the Traveling Salesman Problem and a few other NP-hard problems by combining in a novel way quantum search with classical dynamic programming. In this paper, we show how to apply this approach to the minimum Steiner tree problem, a well-known NP-hard problem, and construct the first quantum algorithm that solves this problem faster than the best known classical algorithms. More precisely, the complexity of our quantum algorithm is $\mathcal{O}(1.812^k\poly(n))$, where $n$ denotes the number of vertices in the graph and $k$ denotes the number of terminals. In comparison, the best known classical algorithm has complexity $\mathcal{O}(2^k\poly(n))$.