We present a combinatorial proof of the $q$-Pfaff--Saalsch\"utz identity by a composition of explicit bijections, in which $q$-binomial coefficients are interpreted as counting subspaces of $\mathbb{ F}_q$-vector spaces. As a corollary, we obtain a new multiplication rule for quantum binomial coefficients and hence a new presentation of Lusztig's integral form $\mathcal{ U}_{ \mathbb{ Z}[q, q^{ -1}]}(\mathfrak{ sl}_2)$ of the Cartan subalgebra of the quantum group $\mathcal{ U}_q(\mathfrak{ sl}_2)$.
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