WASHINGTON — President Trump began a new era of diplomacy with Russia on Saturday as he and President Vladimir V. Putin conducted an hourlong telephone call, and vowed to repair relations between the countries after nearly three years of conflict that threatened a new Cold War between East and West.

The two leaders discussed fighting terrorism and expanding economic ties, but barely mentioned the wedge that has been driven between Washington and Moscow since Russia annexed Crimea and sponsored a separatist war in eastern Ukraine in 2014. Still, although Mr. Trump had previously expressed a willingness to lift sanctions against Russia, the issue did not come up, according to officials on both sides.

The tone of the conversation was reported to be warm, indicating a drastic shift after relations had broken down between Mr. Putin and former President Barack Obama. “The positive call was a significant start to improving the relationship between the United States and Russia that is in need of repair,” the Trump administration said in a statement. “Both President Trump and President Putin are hopeful that after today’s call, the two sides can move quickly to tackle terrorism and other important issues of mutual concern.”

In its statement, the Kremlin said: “Donald Trump asked to convey a desire for happiness and prosperity for the Russian people, noting that the people in America relate with sympathy to Russia and its citizens.” Mr. Putin answered that Russians feel the same way about Americans, the statement said. Neither side mentioned the Russian hacking of the American election in their statements.