The agreement showcases Russia’s new clout as an oil producer and the importance of its alliance with de facto OPEC leader Saudi Arabia. Cooperation between a Russia-led group of non-OPEC oil nations and the Saudis helped producers prop up prices with deep production cuts in 2017.

Not having Russia and other non-OPEC participants “around the table would be a futile exercise,” Nigerian oil minister Emmanuel Ibe Kachikwu said after the hard-fought agreement had been struck.

But the alliance has seen Saudi Arabia cede some of its market power to Russia. “OPEC, or more precisely Saudi Arabia, has been the head honcho of the oil world for nearly six decades. Yet these days it seems unable to make a decision without Russia’s blessing, let alone without risking the wrath of the U.S. president,” said Stephen Brennock, analyst at brokerage PVM Oil Associates Ltd.

Talks had hit a snag earlier this week when Saudi Arabia and Russia disagreed about sharing the pain of curbs to stop a 30% drop in oil prices. The sway held by Russia over the cartel’s decisions has suggested that Saudi Arabia was losing its grip on the group, delegates said. Although Russia isn’t an OPEC member, the Saudis have insisted on Moscow’s joining recent meetings.

As part of the final agreement, Russia will curb its output by 230,000 barrels a day, slightly less than Saudi Arabia, which is reducing by a daily 250,000 barrels. OPEC granted exemptions to Iran, Libya, Venezuela and Nigeria, which had argued to be left out of the cuts due to economic strife in their countries.

The OPEC-Russia partnership is “now like a common-law marriage,” said an OPEC official. “They can’t agree without Saudi Arabia and Russia. And Russia is a strong partner.”

The two nations began negotiations this week with a disagreement over sharing the pain. They clashed when Russia proposed to cut as little as 100,000 barrels a day, according to people familiar with the matter.

That would have meant Saudi Arabia would need to reduce its output five times more than Russia, despite both pumping an average of over 10 million barrels a day. Saudi energy minister Khalid al-Falih told his Russian counterpart, Alexander Novak, to inform Russian President Vladimir Putin that the offer was unacceptable, one person said.

After clashing with Mr. Falih, Mr. Novak flew back to Russia and consulted Mr. Putin, according to a Russian official. When Mr. Novak returned to Vienna on Friday, he met Mr. Falih, who skipped two hours of the OPEC gathering, to haggle with him. After the two sides consulted their respective governments over the phone, an OPEC official said a relaxed Mr. Falih re-entered the OPEC meeting room.

He had a deal with Russia. The non-OPEC pact still had to be struck, and OPEC members also had to reach an agreement, which ultimately led to the exemption of four nations from the curbs.

The negotiations took place in the midst of crisis within the nearly 60-year-old organization. Some smaller oil-producing nations are worried that their voices have been diminished in the two years since the cartel formed an alliance with Russia. Qatar, one of OPEC’s smallest oil-producers, said this week it was severing ties with the group.

The Riyadh-Moscow relationship is a key reason behind Qatar’s surprise exit from OPEC after 57 years with the cartel, an official in the emirate said.

Some OPEC members have also worried that Saudi Arabia is too beholden to U.S. President Donald Trump after the White House stood by the kingdom despite the killing of a journalist by Saudi operatives.

Mr. Falih said the final decision on output wasn’t driven by any political agenda. “President Trump has announced his energy policy—he wants oil-and-gas industry in U.S. to grow—hopefully we can grow together, by keeping markets on a moderate track going forward,” he said.

Write to Benoit Faucon at benoit.faucon@wsj.com, Summer Said at summer.said@wsj.com and Christopher Alessi at christopher.alessi@wsj.com