Let us consider the differential equation $$ \dot{x}=(A+\varepsilon Q(t,\varepsilon))x, \;\;\;\; |\varepsilon|\le\varepsilon_0, $$ where $A$ is an elliptic constant matrix and $Q$ depends on time in a quasiperiodic (and analytic) way. It is also assumed that the eigenvalues of $A$ and the basic frequencies of $Q$ satisfy a diophantine condition. Then it is proved that this system can be reduced to $$ \dot{y}=(A^{*}(\varepsilon)+\varepsilon R^{*}(t,\varepsilon))y, \;\;\;\; |\varepsilon|\le\varepsilon_0, $$ where $R^{*}$ is exponentially small in $\varepsilon$, and the linear change of variables that performs such reduction is also quasiperiodic with the same basic frequencies than $Q$. The results are illustrated and discussed in a practical example.