We prove that the generating function of partitions into $k$-th powers is strongly Gaussian in the sense of Báez-Duarte. Within the probabilistic framework of Khinchin families, the Hardy--Ramanujan asymptotic formula for the number~$p_k(n)$ of partitions of~$n$ into $k$-th powers reads \[ p_k(n) \sim \frac{α_k}{n^{(3k+1)/(2k+2)}} \exp\bigl(β_k\, n^{1/(k+1)}\bigr), \qquad n \to \infty, \] where $α_k$ and $β_k$ are explicit constants depending only on~$k$, then follows directly from Hayman's asymptotic formula for strongly Gaussian power series. The proof of strong Gaussianity combines a Gaussianity criterion for Khinchin families with bounds of Tenenbaum, Wu and Li on the generating function; the asymptotic formula is recovered by computing asymptotic approximations of the mean and variance of the associated family.