In this article we establish a new formula for the difference of a test function of the solution of a stochastic differential equation and of the test function of an Itô process. The introduced formula essentially generalizes both the classical Alekseev-Gröbner formula from the literature on deterministic differential equations as well as the classical Itô formula from stochastic analysis. The discovered formula, which we suggest to refer to as Itô-Alekseev-Gröbner formula, is a powerful tool for deriving strong approximation rates for perturbations and approximations of stochastic ordinary and partial differential equations.