Remember indeed that the solution of a -th order linear differential equation is a component of the solution of a -dimensional linear equation. Then an exponential bound on the euclidean norm of gives an exponential bound on . Here's how to obtain such a bound.
If where are -matrices, we have for every , so that , hence . And now you can apply the one-dimensional Gronwall lemma "in integral form" (quoting the wikipedia) to the function . The only online reference I could find for the final result is this french exercise sheet (Exercice 1.4), but you don't need it to find the proof, just apply Gronwall's lemma like I wrote and do an integration by parts to compute the integral. You should get:
This is what you need.