Trump and Abe also discussed an upcoming visit to Japan and other countries in the region by newly installed Defense Secretary James Mattis. Abe, who during Trump's transition phase became the first foreign leader to talk face-to-face with the president-elect, agreed to meet Trump during a visit to Washington on February 10, according to the White House.
Trump then spoke by phone with German Chancellor Angela Merkel. His outreach to Merkel comes after his repeated attacks on her during the campaign, during which he blasted the German policy on admitting Syrian refugees for allegedly putting German citizens in danger of terrorist attacks.
Trump is planning to speak later Saturday with French President François Hollande and Australian Prime Minister Malcolm Turnbull.
Ahead of Trump's call with Putin, leaders in Moscow expressed cautious hope that the new American leader could forge stronger ties than former president Barack Obama did.
On Saturday, Nikolai Patrushev, the influential head of the Russian Security Council, welcomed the first contact.
"We will await the results, but I believe everything will be positive," Patrushev said Saturday, according to Russia's Interfax news agency.
From Moscow's point of view, lifting the sanctions imposed by the Obama administration for interference in the presidential election and Russia's intervention in Ukraine would be a good start, as would a reduction of NATO's military presence near Russia's borders.
Washington's European allies, meanwhile, have expressed concern over whether Trump's first moves with Russia will signal a reduction of the US commitment to European security.
But Trump, speaking Friday at a White House news conference with British Prime Minister Theresa May, said that it is "very early" to discuss lifting sanctions on Russia. May also stated her commitment to keep the sanctions in place until the Minsk Agreement, a plan to end the conflict in Ukraine, has been implemented. And she added that she continues to argue that position "inside the European Union."
Trump's first contact with Putin as president comes after months of speculation over the Kremlin's role in the 2016 election - starting with Trump's frequent expressions of admiration for Putin and culminating in the assessment of the U.S. intelligence community that Russia interfered in the campaign on Trump's behalf.
Trump has vehemently denied allegations that his positive view of Moscow stems from business ties or blackmail by Russian intelligence, and he has sought to portray his upbeat words about Putin as a positive.
He has consistently argued that Russia can be a strong ally instead of a strategic ally, saying the two countries could cooperate on counterterrorism in general and rolling back the Islamic State in particular, as well as countering nuclear weapon proliferation. Trump has suggested that Washington can work with Moscow on the conflicts in Syria and Ukraine and that he might be ready to negotiate down NATO's strong defensive posture on Russia's western border.
US lawmakers from both parties, and others including Trump Cabinet picks, have raised alarms or at least questioned his softer approach to Russia.
But on Friday, the president expressed more tempered expectations.
"As far as, again, Putin and Russia, I don't say good, bad or indifferent," Trump said. "I don't know the gentleman. I hope we have a fantastic relationship. That's possible. And it's also possible we won't. We will see what happens. I will be representing the American people very, very strongly, very forcefully."
On a grander scale, the Kremlin seems to hope the Trump administration will relax what it sees as a policy of containment since the fall of the Soviet Union left the United States as the world's sole superpower. In the new world order outlined by Putin, Russia would have greater influence in world affairs and, from Moscow's point of view, feel more secure at home.
But Moscow has consistently cautioned about "excessive optimism" over what Trump's presidency will mean for Russia, and Putin spokesman Dmitry Peskov stayed on script Friday. "One can hardly expect substantive contacts on the entire range of affairs from this call," he told reporters. "Let us wait and see. Let us be patient."
Moscow's establishment has welcomed Trump as a pragmatist who will not try to enforce American values on the rest of the world.
"He is a businessman. He is a pragmatic person," Andrei Norkin, co-host of a popular Russian political talk show, said this week. "I hope that his attitude to foreign policy will be like to some sort of business deal. People who will work with him will be telling him 'Mr. President, we are taking a risk here,' and he will agree."
Filipov reported from Moscow. Karen DeYoung in Washington contributed to this report.
