A small Norwegian biotechnology firm said Monday it had the first evidence of real promise in a new approach widely considered to be a potential cure for HIV, prompting both cautious optimism and skepticism from experts.

Eradicating HIV has long proven elusive because the virus lies dormant in so-called latent reservoirs deep in the body. That means that although current treatments have transformed the AIDS-causing virus from a deadly disease to a chronic condition by keeping the virus at bay in the bloodstream, they can’t...