We consider the approximation of systems of reaction-diffusion equations, with the finite element method. The highest derivative in each equation is multiplied by a parameter $\varepsilon \in (0,1]$, and as $\varepsilon \rightarrow 0$ the solution of the system will contain boundary layers. We extend the analysis of the corresponding scalar problem from [Melenk, IMA J. Numer. Anal. 17(1997), pp. 577-601], to construct a finite element scheme which includes elements of size $\mathcal{O}(\varepsilon p)$ near the boundary, where $p$ is the degree of the approximating polynomials. We show that, under the assumption of analytic input data, the method yields exponential rates of convergence, independently of $\varepsilon $, when the error is measured in the energy norm associated with the problem. Numerical computations supporting the theory are also presented, which also show that the method yields robust exponential convergence rates when the error in the maximum norm is used.