We study the full time-domain evolution of gravitational perturbations in black hole spacetimes arising in Einstein-Weyl gravity, a renormalizable extension of general relativity containing quadratic curvature corrections. We analyze both Schwarzschild and non-Schwarzschild solutions, focusing on monopole and higher multipole perturbations. Using semianalytical methods based on the Rezzolla-Zhidenko parametrization for approximation of the black hole spacetime and time-domain integration for analysis of evolution of perturbations, we study the late-time behavior of gravitational perturbations. Our results show that the ringdown phase is followed by universal slowly decaying oscillatory tails with the envelope psi proportional to t-5/6. We also demonstrate the breakdown of the eikonal correspondence between quasinormal modes and unstable null geodesics, highlighting limitations of the Wentzel-Kramers-Brillouin (WKB) method in this context. Our analysis confirms the range of (in)stability of black holes in Einstein-Weyl gravity found in recent publications.