The leptonic decay-constant ratio $f_K/f_\pi$ is calculated from lattice-QCD simulations using $N_f=2+1$ dynamical fermion flavors in the clover-improved formulation and 2-HEX smearing. The simulations were performed for a range of mass-degenerate light quarks including the physical point and at various lattice couplings and volumes, allowing to quantify all relevant sources of systematic uncertainties for our final number of the decay-constant ratio. Utilizing input from chiral perturbation theory, we also quote the charged decay-constant ratio $f_{K^\pm}/f_{\pi^\pm}$. With further input from super-allowed nuclear $\beta$-decays, eventually we obtain an estimate for the CKM-matrix element $V_{\rm us}$.