When President-elect Donald J. Trump’s Middle East envoy, Steve Witkoff, met with Prime Minister Benjamin Netanyahu of Israel on Saturday to pressure him on a cease-fire deal in Gaza, there was someone on the speakerphone: Brett H. McGurk, President Biden’s longtime Mideast negotiator.

It was a vivid example of cooperation between two men representing bitter political rivals whose relationship has been best described as poisonous. Rarely if ever have teams of current and new presidents of different parties worked together at such a high-stakes moment, with the fate of American lives and the future of a devastating war hanging in the balance.

Both Mr. Trump and Mr. Biden publicly claimed credit for the breakthrough.

“This EPIC ceasefire agreement could have only happened as a result of our Historic Victory in November,” Mr. Trump wrote on his social media site even before the deal was formally announced in the Middle East.

At the White House, Mr. Biden told reporters that his administration had worked tirelessly for months to convince the two sides to halt the fighting. He called it “one of the toughest negotiations I’ve ever experienced” and gave credit to “an extraordinary team of American diplomats who have worked nonstop for months to get this done.”

As he left the room, a reporter asked Mr. Biden, “Who gets credit for this, Mr. President, you or Trump?” Mr. Biden stopped, turned around and smiled.

“Is that a joke?” he asked.

But despite the tension between the current president and the next one, their representatives in the Middle East described a cooperative working relationship in the weeks since Election Day.

“Brett is in the lead,” Mr. Witkoff said last week at Mar-a-Lago, Mr. Trump’s club in Florida, describing the working relationship. That description was accurate by all accounts, even if it did not match what Mr. Trump had said moments before in one of several statements describing his negotiators as critical players.

In fact, Mr. Trump’s threat that “all hell” would break loose if no deal was reached before his inauguration on Monday might have helped motivate Hamas’s leadership to make final decisions. But people familiar with the negotiations said the announcement on Wednesday of a deal to temporarily end hostilities in Gaza was the result of months of work by Mr. McGurk in the Middle East, capped off by several weeks of carefully coordinated efforts by Mr. Witkoff.

Mr. Witkoff, 67, a blunt real estate investor from the Bronx, has largely planted himself in Qatar for the negotiations, knowing that whatever Mr. McGurk negotiated, he would have to execute. In fact, the 33 hostages who will be released under the cease-fire deal may not see freedom until Inauguration Day or after. The cease-fire would expire six weeks later, unless Phase 2 of the agreement kicks in.

By design, the goal was to send a unified message that the fighting must end and the hostages held by Hamas must be released. One person familiar with the negotiations, who like others spoke on the condition of anonymity to describe the discussions, said Mr. McGurk was more involved in hammering out details of the agreement, while Mr. Witkoff’s role was to make clear that Mr. Trump wanted a deal by the time he is inaugurated.

The president-elect has also been setting some early parameters in his dealings with Mr. Netanyahu — who, for all his support of Mr. Trump in the election, was perceived by the Trump camp as dragging his feet on a deal. Mr. Witkoff flew to to Israel from Doha on Saturday — despite the Sabbath — to underscore the message that Mr. Netanyahu had to get on board.

Mr. Witkoff’s work, including the meeting with Mr. Netanyahu, helped Mr. McGurk and the Biden administration to put pressure on both sides during the negotiation, according to the person familiar with the talks.

It was not at all clear that such an arrangement would work in the days immediately after Mr. Trump won a second term.

He and Mr. Biden have barely talked in recent weeks, their already acrimonious relationship weighed down by the Trump team’s determination to clean out the White House career staff and the Biden team issuing last-minute orders to box in the new administration.

In his remarks on Wednesday, Mr. Biden acknowledged some level of cooperation and respect between their aides.

“This deal was developed and negotiated under my administration, but its terms will be implemented for the most part by the next administration,” Mr. Biden told reporters. “In these past few days, we’ve been speaking as one team.”

But he did not give any more credit to Mr. Trump for helping the effort. For his part, the president-elect said he was “thrilled” that the American hostages would be released, but he did not mention Mr. Biden or the work of the current administration.

“We have achieved so much without even being in the White House,” Mr. Trump wrote. “Just imagine all of the wonderful things that will happen when I return to the White House, and my Administration is fully confirmed, so they can secure more Victories for the United States!”

Both leaders left it to staff members to describe the way they had worked together on the Gaza negotiations.

A person familiar with that effort said a close partnership between Mr. McGurk and Mr. Witkoff was part of an “incredibly effective” process by which the Biden administration finalized a deal that the Trump administration would have to oversee.

That cooperation began soon after Mr. Trump won the election and named Mr. Witkoff to be his envoy to the region. Biden administration officials have said they believe the momentum for a deal began before that, when Mr. Biden helped broker a separate agreement to end fighting between Israel and Hezbollah in Lebanon. That isolated Hamas and helped persuade the group that a cease-fire was in its interests, according to Biden officials.