The detection of Ly$\alpha$ nebulae around $z\gtrsim 6$ quasars provides evidence for extended gas reservoirs around the first rapidly growing supermassive black holes. Observations of $z > 6$ quasars can be explained by cosmological models provided that the black holes by which they are powered evolve in rare, massive dark matter haloes. Whether these theoretical models also explain the observed extended Ly$\alpha$ emission remains an open question. We post-process a suite of cosmological, radiation-hydrodynamic simulations targeting a quasar host halo at $z>6$ with the Ly$\alpha$ radiative transfer code RASCAS. A combination of recombination radiation from photo-ionised hydrogen and emission from collisionally excited gas powers Ly$\alpha$ nebulae with a surface brightness profile in close agreement with observations. We also find that, even on its own, resonant scattering of the Ly$\alpha$ line associated to the quasar's broad line region can also generate Ly$\alpha$ emission on $\sim 100 \, \rm kpc$ scales, resulting in comparable agreement with observed surface brightness profiles. Even if powered by a broad quasar Ly$\alpha$ line, Ly$\alpha$ nebulae can have narrow line-widths $\lesssim 1000 \, \rm km \, s^{-1}$, consistent with observational constraints. Even if there is no quasar, we find that halo gas cooling produces a faint, extended Ly$\alpha$ glow. However, to light-up extended Ly$\alpha$ nebulae with properties in line with observations, our simulations unambiguously require quasar-powered outflows to clear out the galactic nucleus and allow the Ly$\alpha$ flux to escape and still remain resonant with halo gas. The close match between observations and simulations with quasar outflows suggests that AGN feedback already operates before $z \, = \, 6$ and confirms that high-$z$ quasars reside in massive haloes tracing overdensities.