We explore the possibility that relativistic protons in the extremely powerful jets of blazars may boost via elastic collisions the dark matter particles in the surroundings of the source to high energies. We concentrate on two sample blazars, TXS 0506+056, towards which IceCube recently reported evidence for a high-energy neutrino flux, and BL Lacertae, a representative nearby blazar. We find that the dark matter flux at Earth induced by these sources may be sizable, larger than the flux associated with the analogous process of dark matter boosted by galactic cosmic rays, and relevant to access direct detection for dark matter particle masses lighter than 1 GeV. From the null detection of a signal by XENON1T, MiniBooNE, and Borexino, we derive limits on dark matter-nucleus spin-independent and spin-dependent cross sections which, depending on the modelization of the source, improve on other currently available bounds for light dark matter candidates of 1 up to 5 orders of magnitude.