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We study the differentially private top-$k$ selection problem, aiming to identify a sequence of $k$ items with approximately the highest scores from $d$ items. Recent work by Gillenwater et al. (2022) employs a direct sampling approach from the vast collection of $O(d^k)$ possible length-$k$ sequences, showing superior empirical accuracy compared to previous pure or approximate differentially private methods. Their algorithm has a time and space complexity of $\tilde{O}(dk)$. In this paper, we present an improved algorithm that achieves time and space complexity of $\tilde{O}(d + k^2)$. Experimental results show that our algorithm runs orders of magnitude faster than their approach, while achieving similar empirical accuracy.