Yes. For nonnegative (as opposed to strictly positive) coefficients, a more general result is available (in several variables). See the article by me on powers of polynomials in Symbolic Dynamics and Its Applications (Editors, Adler and Walters, AMS 1992 (unfortunately, I can't find a link to it) [I have three articles there]. In several variables, the formulation is, if $P$ is a polynomial such that $P(1,1,\dots,1)> 0$, and there exists $m$ such that $P^m$ has only nonnegative coefficients, then for all sufficiently large $M$, $P^M$ has only nonnegative coefficients.
In one variable, the strict positivity condition you require can also be dealt with from this, but is a nonstarter in more than two variables. Note that if the second largest or the second smallest degree monomial has zero coefficient, then that persists in all powers, so no power of such poly in one variable can be strictly positive in your sense. However, the answer to your question is yes, because the gaps will eventually fill in ... (this requires a little, but not much, work).
Edit: Here is the reference (found, finally, in Zentralblatt): Handelman, David,
Polynomials with a positive power. (Zbl 0794.26013)
Symbolic dynamics and its applications, Proc. AMS Conf. in honor of R. L. Adler, New Haven/CT (USA) 1991, Contemp. Math. 135, 229-230 (1992).
[For the entire collection see Zbl 0755.00019.]