There is a positive answer to this question now by Pham Hung Quy's post here and this paper, but let me also mention that the same thing is not true if the $\mathbb{Z}$-grading is changed to a $Q$-grading where $Q$ is an arbitrary commutative Noetherian monoid.
By a result of Eisenbud and Sturmfels (Binomial ideals, Proposition 1.11), $Q$-graded ideals are exactly binomial ideals. There are binomial ideals that are reducible as ideals, but cannot be written as intersections of binomial ideals. My paper with Ezra Miller and Chris O'Neill gives the first example: In $k[x,y]$, the ideal $(x^2y-xy^2, x^3, y^3)$ has two irreducible components (because its socle is two dimensional), and one of them must contain a non-binomial. See Example 6.1 and Theorem 6.4 in the linked paper.