HAMBURG—Coming face-to-face in a highly anticipated meeting, the American and Russian presidents disagreed Friday over election interference and about the best approach to North Korea, but made tentative progress toward curbing the bloodshed in Syria’s long-running war.

President Donald Trump and Russia’s Vladimir Putin spoke for more than two hours Friday on the sidelines of the Group of 20 leading nations summit in Germany. The meeting went so much longer than planned that first lady Melania Trump looked in at one point to see if she could coax them to wrap up.

It didn’t work. They kept talking another hour.

“It was an extraordinarily important meeting—so much for us to talk about,” Secretary of State Rex Tillerson, who was in the room, told reporters afterward. “And it was a good start.”

Mr. Trump’s interest lay not only in speaking in detail with the Russian leader, but also in trying to shape the narrative that emerged about the meeting. Toward that end, Mr. Tillerson provided a round-by-round account of the conversation, answering questions from reporters about a spectrum of international issues.