Background: The COVID-19 global health crisis has led to an exponential surge in published scientific literature. In an attempt to tackle the pandemic, extremely large COVID-19-related corpora are being created, sometimes with inaccurate information, which is no longer at scale of human analyses. Objective: In the context of searching for scientific evidence in the deluge of COVID-19-related literature, we present an information retrieval methodology for effective identification of relevant sources to answer biomedical queries posed using natural language. Methods: Our multistage retrieval methodology combines probabilistic weighting models and reranking algorithms based on deep neural architectures to boost the ranking of relevant documents. Similarity of COVID-19 queries is compared to documents, and a series of postprocessing methods is applied to the initial ranking list to improve the match between the query and the biomedical information source and boost the position of relevant documents. Results: The methodology was evaluated in the context of the TREC-COVID challenge, achieving competitive results with the top-ranking teams participating in the competition. Particularly, the combination of bag-of-words and deep neural language models significantly outperformed an Okapi Best Match 25-based baseline, retrieving on average, 83% of relevant documents in the top 20. Conclusions: These results indicate that multistage retrieval supported by deep learning could enhance identification of literature for COVID-19-related questions posed using natural language.