Write $V=k^n$ and $H={\rm GL}(V)$.
 Let $T(V)$ denote the tensor algebra of $V$, and let $\theta$ denote the (infinite dimensional) representation of $H$ in $T(V)$.
Since the unipotent group $G\subset H$ has no nontrivial characters, it is the stabilizer of some tensor $t_0\in T(V)$; see any of the books titled "Linear Algebraic Groups". The Lie
algebra ${\rm Lie}(H)$ acts in $T(V)$ via $d\theta$, and 
$$ {\rm Lie}(G)=\{X\in {\rm Lie}(H)\ |\  (d\theta)(X)(t_0)=0\}. $$
Now for the representation $\theta$ in $T(V)$ we have 
$$ \theta(\exp(X))=\exp((d\theta)(X))$$
for $X\in {\rm Lie}(H)$ (this is an assertion about formal power series in $X$).
Since for $X\in {\rm Lie}(G)$ we have $(d\theta)(X)(t_0)=0$, we conclude that $\theta(\exp(X))(t_0)=t_0$,  that is, $\exp(X)\in G(k)$, as required.