We show that any SDE driven by Brownian motion with drift satisfying the Krylov–Röckner condition has exactly one solution in an ordinary sense for almost every trajectory of the Brownian motion, i.e. path-by-path uniqueness holds. Additionally, we show that such SDE is strongly complete, i.e. for almost every trajectory of the Brownian motion, the family of solutions with different initial data forms a continuous semiflow for all nonnegative times.