We present a fairly general framework for reducing $(\varepsilon, δ)$ differentially private (DP) statistical estimation to its non-private counterpart. As the main application of this framework, we give a polynomial time and $(\varepsilon,δ)$-DP algorithm for learning (unrestricted) Gaussian distributions in $\mathbb{R}^d$. The sample complexity of our approach for learning the Gaussian up to total variation distance $α$ is $\widetilde{O}(d^2/α^2 + d^2\sqrt{\ln(1/δ)}/α\varepsilon + d\ln(1/δ) / α\varepsilon)$ matching (up to logarithmic factors) the best known information-theoretic (non-efficient) sample complexity upper bound due to Aden-Ali, Ashtiani, and Kamath (ALT'21). In an independent work, Kamath, Mouzakis, Singhal, Steinke, and Ullman (arXiv:2111.04609) proved a similar result using a different approach and with $O(d^{5/2})$ sample complexity dependence on $d$. As another application of our framework, we provide the first polynomial time $(\varepsilon, δ)$-DP algorithm for robust learning of (unrestricted) Gaussians with sample complexity $\widetilde{O}(d^{3.5})$. In another independent work, Kothari, Manurangsi, and Velingker (arXiv:2112.03548) also provided a polynomial time $(\varepsilon, δ)$-DP algorithm for robust learning of Gaussians with sample complexity $\widetilde{O}(d^8)$.