Mr. Trump seemed to presage a different era with all the praise he heaped on Russia and Mr. Putin. He described him as a strong, smart leader and said that Moscow seemed to be blamed for everything. And he called for better relations with Moscow to fight the Islamic State and other terrorist groups, echoing a longstanding Putin pitch.

Some voices in Moscow cautioned that Mrs. Clinton, as a calmer hand on the tiller, would be the kind of predictable leader that the Kremlin preferred, albeit a hostile one. Now, there is a sense that the Kremlin might be unsettled by the president of a far more powerful country deploying Mr. Putin’s favorite tactic: unpredictability.

“Trump will be tamed and act more presidential, eventually, but he also has a penchant for unpredictability that works against the Kremlin,” said Konstantin von Eggert, a political commentator for TV Rain, Russia’s only independent channel. “This creates a situation in which a stronger player with the same style of unpredictability as a strategy comes on the stage. Putin did not anticipate that.”

There has been a certain amount of policy whiplash on issues important to Russia. First, Mr. Trump said that NATO was obsolete, then that it had America’s solid backing. He seemed to indicate he would lift economic sanctions imposed over the Ukraine crisis, and appointed as secretary of state Rex W. Tillerson, who as head of Exxon Mobil cut enormous oil deals with Russia and spoke out publicly against sanctions.

Then the new United States ambassador to the United Nations, Nikki R. Haley, sharply criticized Russia over Ukraine, suggesting that sanctions were hinged to a peace deal there. Mr. Tillerson echoed that line.

Finally, Mr. Trump started to mix geopolitical apples and oranges, crossing issues in a way that Moscow deplores. He said maybe sanctions could be lifted in exchange for a better deal on nuclear arms. The Trump administration seemed to want the Kremlin to distance itself from Iran, its ally in Syria, and from China.

“There is a cautious feeling about how Trump and his advisers designated the possible ways of improving relations with Russia,” said Vladimir Frolov, an international affairs analyst. “This has frightened the Kremlin because it does not correspond to Russia’s interests.”