Let $L = −∆+V$ be a Schrödinger operator acting on $L^2(\mathbb{R}^n)$, $n ≥ 1$, where $V \not\equiv 0$ is a nonnegative locally integrable function on $\mathbb{R}^n$. In this article, we will introduce weighted Hardy spaces $H^p_L(w)$ associated with $L$ by means of the square function and then study their atomic decomposition theory. We will also show that the Riesz transform $∇L^{−1/2}$ associated with $L$ is bounded from our new space $H^p_L (w)$ to the classical weighted Hardy space $H^p(w)$ when $n/(n+1)< p<1$ and $w ∈ A_1∩RH_{(2/p)′}$.