That mandate made sense when time was on Ukraine’s side, one of Mr. Biden’s top advisers said over the weekend. But now, momentum has reversed. Mr. Zelensky, who clashed repeatedly with Mr. Biden and his staff over their reluctance to give him long-range artillery, then tanks, then F-16’s, began a public pressure campaign to get Mr. Biden to soften his restrictions on firing American weapons across the border into Russia.

In an interview with The New York Times two weeks ago, Mr. Zelensky addressed Mr. Biden.

“Shoot down what’s in the sky over Ukraine,” Mr. Zelensky said. “And give us the weapons to use against Russian forces on the borders.” He was taking public what he had been saying more insistently to visiting American officials, including Secretary of State Antony J. Blinken, the most recent senior official to visit Kyiv.

Mr. Blinken came back convinced, and in a Friday night Oval Office meeting with Mr. Biden, he and Jake Sullivan, the national security adviser, along with Defense Secretary Lloyd J. Austin III, convinced Mr. Biden that he needed to lift the restrictions at least in the border areas around Kharkiv, Ukraine’s second-largest city. Otherwise, they warned, Russia could well start taking back significant parts of territory that it was chased out of in the fall of 2022.

Ukraine announced on Monday that it had used Western-provided weapons to destroy an air-defense system on Russian territory, though it did not name the weapon or give details. Russia’s deputy foreign minister, Sergei A. Ryabkov, then issued a warning that if Western-provided weapons struck Russia, Moscow would extract “fatal consequences.”

Mr. Biden’s aides insisted that the president had not reversed himself as much as he had created an exception to his non-escalation rule. But as Mr. Blinken himself hinted at the end of last week, it might not be the last exception. He said the American strategy to push back on Russia would adapt to changes on the battlefield.