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.