Large language models require vast amounts of high-quality training data, but effective filtering of web-scale datasets remains a significant challenge. This paper demonstrates that GPT-4o is remarkably effective at identifying high-quality training data, but its prohibitive cost makes it impractical at web-scale. We propose SIEVE, a lightweight alternative that matches GPT-4o accuracy at less than 1\% of the cost. SIEVE can perform up to 500 filtering operations for the cost of one GPT-4o filtering call. The key to SIEVE is a seamless integration of GPT-4o and lightweight text classification models, using active learning to fine-tune these models in the background with a small number of calls to GPT-4o. Once trained, it performs as well as GPT-4o at a tiny fraction of the cost. Through different filtering prompts, SIEVE can efficiently curate high quality data for general or specialized domains from web-scale corpora -- a valuable capability given the current scarcity of high-quality domain-specific datasets. Extensive experiments using automatic and human evaluation metrics show that SIEVE and GPT-4o achieve similar performance on five highly specific filtering prompts. In addition, when performing quality filtering on web crawl datasets, we demonstrate SIEVE can further improve over state-of-the-art quality filtering methods in the DataComp-LM challenge for selecting LLM pretraining data.