“They have to strictly observe their promises” made in the treaty, Mr. Abe said after the 45-minute meeting. “I called on the Korean side to come up with the catalyst for restoring Japan-South Korea relations to health.”

Mr. Moon’s spokeswoman said disagreements remained on the forced-labor issue but the two leaders agreed on the need to quickly resolve the issue and gather for additional summits. Mr. Moon repeated to Mr. Abe that he couldn't tamper with court decisions, she said.

The two leaders met in Chengdu, the capital of western China’s Sichuan province, after a three-way summit that also included Chinese Premier Li Keqiang.

Over the summer, Japan imposed export restrictions on materials needed by South Korean technology companies, prompting Mr. Moon to announce plans to withdraw from a military information-sharing agreement with Tokyo.

Mr. Moon recently took a move toward reconciliation by saying South Korea would stay in the military agreement after all. And at the opening of the meeting, he congratulated Mr. Abe on becoming the longest-serving prime minister in Japanese history earlier this year.

The Japanese leader offered no in-kind gesture. During the meeting, Mr. Abe said relations were in an “extremely severe state” and said the reason boiled down to the forced-labor issue, according to Naoki Okada, a Japanese official who was present. The atmosphere of the meeting was “tense, but not acrimonious,” Mr. Okada said.

In 2020, South Korea’s Supreme Court may proceed with plans to liquidate assets seized from the Japanese companies targeted in the earlier court ruling about forced labor. If so, “it would mean that the two sides are really turning their backs to each other,” said Shin Beom-chul, a senior researcher at the Asan Policy Institute, a private think tank in Seoul.

Analysts said Mr. Moon’s chief objective at the meeting was to ease Tokyo’s export controls, a priority that differed from Mr. Abe’s. Mr. Moon praised Tokyo’s recent move to open talks over the issue and suggested he was looking for further progress. The Japanese side said it didn’t take him up on the subject.

Last week, Tokyo lifted controls on one of the materials, but it has resisted linking the issue to other disputes between the countries.

Although the Trump administration generally hasn’t sought to mediate a deal between Tokyo and Seoul over their disputes on history and trade, the bad blood is worrisome to Washington, since the two countries both host U.S. troops and are allies in trying to hold back North Korea’s nuclear program.

Mr. Moon alluded to U.S. concerns at the start of his meeting with Mr. Abe, which was open to reporters. He said the rare meeting had drawn global attention, “and we understand what the expectations are.”

Mr. Abe denounced recent missile launches by North Korea, but his criticism wasn’t fully supported by either South Korea or China during their three-way meeting. The three issued a joint statement that held off from making direct demands on Pyongyang.

Meeting Mr. Abe in Beijing on Monday, President Xi Jinping broached a proposal by China and Russia to lift some sanctions on North Korea—a position at odds with Japan.

—Andrew Jeong in Seoul contributed to this article.

Write to Peter Landers at peter.landers@wsj.com