We construct an injective algebra homomorphism of the quantum group $U_q(mathfrak{sl}_{n+1})$ into a quantum cluster algebra $mathbf{L}_n$ associated to the moduli space of framed $PGL_{n+1}$-local systems on a marked punctured disk. We obtain a description of the coproduct of $U_q(mathfrak{sl}_{n+1})$ in terms of the corresponding quantum cluster algebra associated to the marked twice punctured disk, and express the action of the $R$-matrix in terms of a mapping class group element corresponding to the half-Dehn twist rotating one puncture about the other. As a consequence, we realize the algebra automorphism of $U_q(mathfrak{sl}_{n+1})^{otimes 2}$ given by conjugation by the $R$-matrix as an explicit sequence of cluster mutations, and derive a refined factorization of the $R$-matrix into quantum dilogarithms of cluster monomials.
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