Screening tasks that aim to identify a small subset of top alternatives from a large pool are common in business decision-making processes. These tasks often require substantial human effort to evaluate each alternative's performance, making them time-consuming and costly. Motivated by recent advances in large language models (LLMs), particularly their ability to generate outputs that align well with human evaluations, we consider an LLM-as-human-evaluator approach for conducting screening virtually, thereby reducing the cost burden. To achieve scalability and cost-effectiveness in virtual screening, we identify that the stochastic nature of LLM outputs and their cost structure necessitate efficient budget allocation across all alternatives. To address this, we propose using a top-$m$ greedy evaluation mechanism, a simple yet effective approach that keeps evaluating the current top-$m$ alternatives, and design the explore-first top-$m$ greedy (EFG-$m$) algorithm. We prove that EFG-$m$ is both sample-optimal and consistent in large-scale virtual screening. Surprisingly, we also uncover a bonus ranking effect, where the algorithm naturally induces an indifference-based ranking within the selected subset. To further enhance practicality, we design a suite of algorithm variants to improve screening performance and computational efficiency. Numerical experiments validate our results and demonstrate the effectiveness of our algorithms. Lastly, we conduct a case study on LLM-based virtual screening. The study shows that while LLMs alone may not provide meaningful screening and ranking results when directly queried, integrating them with our sample-optimal algorithms unlocks their potential for cost-effective, large-scale virtual screening.