In this paper we aim at employing a compactness criterion of Da Prato, Malliavin, Nualart (C. R. Math. Acad. Sci. Paris 315 (1992) 1287–1291) for square integrable Brownian functionals to construct strong solutions of SDE’s under an integrability condition on the drift coefficient. The obtained solutions turn out to be Malliavin differentiable and are used to derive a Bismut–Elworthy–Li formula for solutions of the Kolmogorov equation. We emphasise that our approach exhibits high flexibility to study a variety of other types of stochastic (partial) differential equations as e.g. stochastic differential equations driven by fractional Brownian motion.