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. (ICML '22) employs a direct sampling approach from the vast collection of $d^{\,Θ(k)}$ 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 with time and space complexity $O(d + k^2 / ε\cdot \ln d)$, where $ε$ denotes the privacy parameter. Experimental results show that our algorithm runs orders of magnitude faster than their approach, while achieving similar empirical accuracy.