We investigate quasar outflows at z ≥ 6 by performing zoom-in cosmological hydrodynamical simulations. By employing the smoothed particle hydrodynamics code GADGET-3, we zoom in the 2R200 region around a 2 × 1012M· halo at z = 6, inside a (500 Mpc)3 comoving volume. We compare the results of our active galactic nuclei (AGN) runs with a control simulation in which only stellar/SN feedback is considered. Seeding 105M· black holes (BHs) at the centres of 109M· haloes, we find the following results. BHs accrete gas at the Eddington rate over z = 9-6. At z = 6, our most-massive BH has grown to MBH = 4 × 109M·. Fast (vr > 1000 km s-1), powerful (Mout ~ 2000M· yr-1) outflows of shock-heated low-density gas form at z ~ 7, and propagate up to hundreds kpc. Star formation is quenched over z = 8-6, and the total star formation rate (SFR surface density near the galaxy centre) is reduced by a factor of 5 (1000). We analyse the relative contribution of multiple physical process: (i) disrupting cosmic filamentary cold gas inflows, (ii) reducing central gas density, (iii) ejecting gas outside the galaxy; and find that AGN feedback has the following effects at z = 6. The inflowing gas mass fraction is reduced by~12 per cent, the high-density gas fraction is lowered by ~13 per cent, and ~20 per cent of the gas outflows at a speed larger than the escape velocity (500 km s-1). We conclude that quasar-host galaxies at z ≥ 6 are accreting non-negligible amount of cosmic gas, nevertheless AGN feedback quenches their star formation dominantly by powerful outflows ejecting gas out of the host galaxy halo.This work is supported by the PRIN-INAF 2014 grant ‘Windy black holes combing galaxy evolution’. CC acknowledges funding from the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No. 664931. RM acknowledges support from the Science and Technology Facilities Council (STFC) and from the ERC Advanced Grant 695671 ‘QUENCH’. SC acknowledges financial support from the STFC