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The goal of this article is to establish a central limit theorem for theEuler-Maruyama scheme approximating multidimensional SDEs with ellipticBrownian diffusion, under very mild regularity requirements on the driftcoefficients. When the drift is H\"older continuous, we show that the limitinglaw of the rescaled fluctuations around the true solution is characterised asthe unique solution of a hybrid Young-It\^o differential equation. When thedrift has positive Sobolev regularity, this limit is characterised by thesolution of a transformed SDE. Our result is an extension of the results ofJacod-Kurtz-Protter (1991, 1998) in which SDEs with differentiable coefficientswere considered. To compensate for the lack of regularity of the drifts, weutilize the regularisation effect from the non-degenerate noise.

A central limit theorem for the Euler method for SDEs with irregular drifts