Computational social science (CSS) practitioners often rely on human-labeled data to fine-tune supervised text classifiers. We assess the potential for researchers to augment or replace human-generated training data with surrogate training labels from generative large language models (LLMs). We introduce a recommended workflow and test this LLM application by replicating 14 classification tasks and measuring performance. We employ a novel corpus of English-language text classification data sets from recent CSS articles in high-impact journals. Because these data sets are stored in password-protected archives, our analyses are less prone to issues of contamination. For each task, we compare supervised classifiers fine-tuned using GPT-4 labels against classifiers fine-tuned with human annotations and against labels from GPT-4 and Mistral-7B with few-shot in-context learning. Our findings indicate that supervised classification models fine-tuned on LLM-generated labels perform comparably to models fine-tuned with labels from human annotators. Fine-tuning models using LLM-generated labels can be a fast, efficient and cost-effective method of building supervised text classifiers.