We show that if the row generating functions of an exponential Riordan array $R=[g(t),f(t)]$ have only real non-positive zeros, then so do those of the triangular matrix $R\cdot S_{\sigma}^m$, where $S_{\sigma}=[1,\sigma (e^{t}-1)]$ is an exponential Riordan array with $\sigma\ge 0$ and $m=1,2,\ldots.$ As applications, we derive in a unified approach that row generating functions of many well-known triangular matrices have only real zeros. In particular, we show that the row generating functions $S^k$ have only real zeros for $k=1,2,\ldots,$ where $S$ is the triangle of Stirling numbers of the second kind.