Fix a poset Q on $\{x_1,\ldots ,x_n\}$ . A Q-Borel monomial ideal $I \subseteq \mathbb {K}[x_1,\ldots ,x_n]$ is a monomial ideal whose monomials are closed under the Borel-like moves induced by Q. A monomial ideal I is a principal Q-Borel ideal, denoted $I=Q(m)$ , if there is a monomial m such that all the minimal generators of I can be obtained via Q-Borel moves from m. In this paper we study powers of principal Q-Borel ideals. Among our results, we show that all powers of $Q(m)$ agree with their symbolic powers, and that the ideal $Q(m)$ satisfies the persistence property for associated primes. We also compute the analytic spread of $Q(m)$ in terms of the poset Q.