We define atomic Hardy space $H^p_{\mathcal{L}, at}(\mathbb{C}^n), 0<p\leq 1$ for the twisted Laplacian $\mathcal{L}$ and prove its equivalence with the Hardy space defined using the maximal function corresponding to the heat semigroup $e^{-t\mathcal{L}},t>0$. We also prove sharp $L^p, 0<p\leq 1$ estimates for $\left(\mathcal{L}\right)^{-\beta/2}e^{i\sqrt{\mathcal{L}}}$. More precisely we prove that it is a bounded operator on $H^p_{\mathcal{L}, at}(\mathbb{C}^n)$ when $\beta\geq (2n-1)\left(1/p-1/2\right)$.