We propose a second order exponential scheme suitable for two-component coupled systems of stiff advection--diffusion--reaction equations in two and three space dimensions. It is based on a directional splitting of the involved matrix functions, which allows for a simple yet efficient implementation through the computation of small-sized exponential-like functions and tensor-matrix products. The procedure straightforwardly extends to the case of an arbitrary number of components and to any space dimension $d$. Several numerical experiments in 2D and 3D with physically relevant DIB, Schnakenberg, FitzHugh--Nagumo, and advective Brusselator models clearly show the advantage of the approach against state-of-the-art techniques.
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