We investigate the effects of Dzyaloshinsky-Moriya (DM) interactions on the frustrated J1−J2 kagome-Heisenberg model using the pseudofermion functional renormalization group (PFFRG) technique. In order to treat the off-diagonal nature of DM interactions, we develop an extended PFFRG scheme. We benchmark this approach in parameter regimes that have previously been studied with other methods and find good agreement of the magnetic phase diagram. Particularly, finite DM interactions are found to stabilize all types of noncollinear magnetic orders of the J1−J2 Heisenberg model (q=0, √3×√3, and cuboc orders) and shrink the extents of magnetically disordered phases. We discuss our results in the light of the mineral herbertsmithite which has been experimentally predicted to host a quantum spin liquid at low temperatures. Our PFFRG data indicate that this material lies in close proximity to a quantum critical point. In parts of the experimentally relevant parameter regime for herbertsmithite, the spin-correlation profile is found to be in good qualitative agreement with recent inelastic-neutron-scattering data.