Suppose first that $p'$ is nonzero on $(0,1)$, so that $p$ has an inverse $q$ on $[0,1]$ which is differentiable everywhere except possibly at the endpoints. Then making the substitution $u=p(x)$ yields $\int_0^1 e^{i \, n \, p(x)} \, dx=\int_{p(0)}^{p(1)} e^{i \, n \, u} \, q'(u) \, du \, .$ Since $q'$ is the derivative of a bounded function, it is integrable. So the ordinary Riemann-Lebesgue lemma applies.
In general, $p'$ may not be nonzero on the entire interval. But since $p$ is a polynomial, we can split $[0,1]$ up into some finite collection of intervals over which $p'$ is nonzero, and make the above substitution separately on each of them...