We compute the Chow rings with integral coefficients of moduli stacks of minimal Weierstrass fibrations over the projective line. For each integer \(N\geq 1\), there is a moduli stack \(\mathcal{W}^{\mathrm{min}}_N\) parametrizing minimal Weierstrass fibrations with fundamental invariant \(N\). Following work of Miranda and Park--Schmitt, we give a quotient stack presentation for each \(\mathcal{W}^{\mathrm{min}}_N\). Using these presentations and equivariant intersection theory, we determine a complete set of generators and relations for each of the Chow rings. For the cases \(N=1\) (respectively, \(N=2\)), parametrizing rational (respectively, K3) elliptic surfaces, we give a more explicit computation of the relations.