MOSCOW—With his win in the U.S. presidential election, Donald Trump is uniquely positioned to overhaul America’s relationship with Russia. But how Moscow will engage with the president-elect is still far from clear.

During the campaign, Mr. Trump expressed admiration for Russian President Vladimir Putin, and suggested he would tear up Washington’s Russia playbook and work more closely with the Kremlin. He also said he could meet with Mr. Putin before he is sworn into office.

Mr. Putin on Wednesday congratulated Mr. Trump on his election victory, telling him in a telegram that he hoped they could end a crisis in U.S.-Russian relations and work together on global security and international issues, the Kremlin said.

“We understand that the way forward will be difficult, taking into account the current degraded state of relations between the U.S. and Russia,” Mr. Putin said in separate, televised comments Wednesday. “It’s not our fault that Russian-American relations are in such a poor state. But Russia is prepared and wants to restore full-fledged relations with the United States.”

Russia has faced U.S. and European sanctions and international isolation following the annexation of the Black Sea peninsula of Crimea in 2014 and its backing of separatists in eastern Ukraine.

Earlier this year, Russian officials had expressed hopes that some European states would soften their stance on sanctions, but Moscow’s military support for a Syrian government offensive against the city of Aleppo has infuriated European leaders and raised the prospect of new penalties.

Russia resurfaced as a major foreign-policy challenge for Washington during the campaign. In addition to deploying its military might to challenge U.S. policy in Syria and Ukraine, the Kremlin injected itself into the presidential election by directing the hacking and leaking of emails from the Democratic National Committee, according to U.S. officials. Moscow denies the allegation.

Mr. Putin has also positioned himself as an essential player in resolving several crises that a new U.S. administration will face, said Fiona Hill, senior fellow at the Brookings Institution.

“We’ve basically ceded all the initiative to him, and made Russia a feature of U.S. politics in a way it hasn’t been in a long time,” she said. “Even if he isn’t doing all the things we say he has been doing, we’ve been attributing them to him…and achieving the same result.”

Obama administration officials warn that the Russian-backed hacking effort could continue as the Kremlin tries to influence U.S. politics and elections in Europe next year.

But Mr. Trump has declined to blame Russia for the hacking, and indicated support for policies that could benefit Russia in various ways.

The president-elect has suggested, for example, that his administration could make U.S. support for North Atlantic Treaty Organization members conditional on them spending more on defense; indicated he would consider recognizing Russia’s annexation of Crimea; and said the U.S. should be working with Russia in Syria, where the Kremlin has targeted opponents of the Syrian regime, including rebels backed by the U.S.

“If we got along well, that would be good,” Mr. Trump said in the third presidential debate. “If Russia and the United States got along well and went after ISIS, that would be good,” he said, referring to Islamic State.

Any such large shifts in policy would rock U.S. alliances, particularly in Europe, where there is wariness about Russian intentions.

“A Trump presidency would raise the question of Russia benefiting, not relations necessarily getting better. It might be about weakening the Western power alliance, which would benefit Russia,” said Maria Lipman, a political analyst in Moscow.

It is unclear how quickly Mr. Trump could implement some changes, particularly given bipartisan support in Congress for a strong line against the Kremlin.

The president-elect has discretion to map out a new course in foreign policy. But Mr. Trump’s comments on Russia have found little support from the mainstream foreign-policy establishment, and some Republican foreign-policy lions could intervene to help shape his policies, said Andrew Weiss, vice president at the Carnegie Endowment for International Peace.

Even if Mr. Trump wanted to work with Russia in Syria, that would be unlikely to yield much success because of “a corrosive lack of trust across the board,” said Mr. Weiss, who cited a U.S. airstrike that killed Syrian troops and Russia’s alleged bombing of a United Nations aid convoy.

Mr. Putin’s spokesman, Dmitry Peskov, offered few clues on how Russia would cooperate with a new administration, beyond saying in an interview with Interfax news agency that Mr. Trump’s statements “give hope that, perhaps, we will manage to make some timid steps in the direction of improving our relations.”

He said the Kremlin had no immediate plans for a telephone call or meeting between Mr. Putin and Mr. Trump.