A federal appeals court on Thursday overturned the 2015 corruption conviction of Sheldon Silver, once the powerful speaker of the New York State Assembly, saying the judge’s jury instructions were in error in light of a United States Supreme Court decision that has since narrowed the legal definition of corruption.

Mr. Silver was convicted on charges that he had obtained nearly $4 million in illicit payments in return for taking a series of official actions that benefited others. But in the jury instructions, the judge’s explanation of an official action was too broad, the appeals court found, because it swept in some conduct that the Supreme Court’s decision would now exclude.

Federal prosecutors quickly vowed to retry the case, noting that the appeals court said that the evidence against Mr. Silver was legally sufficient to support a conviction.

The ruling, by a three-judge panel of the United States Court of Appeals for the Second Circuit in Manhattan, was nonetheless a startling development that buttressed Mr. Silver’s reputation as a man whose resilience and influence in New York once seemed boundless.