The swap, which included 24 Ukrainian sailors detained by Russia in the Kerch Strait last year, was likely to be seen as a diplomatic victory both for President Vladimir V. Putin of Russia and for Vladimir A. Zelensky, who ascended to the Ukranian presidency in May.

For Mr. Putin, the swap could open the door to easing tensions with Europe, which imposed sanctions on Russia after the annexation of Crimea in 2014. For the Russian leader — amid an ailing economy and weeks of protests calling for fair municipal elections after opposition candidates were stricken from ballots — the gesture could be a way to demonstrate to Europe that he was willing to compromise.

For Mr. Zelensky, the exchange represented a diplomatic coup and the fulfillment of a significant campaign promise. With his approval ratings high, Mr. Zelensky is aiming to keep his administration’s momentum and deliver results. Defusing tensions in the country’s east was the top priority for Ukrainians, polls showed.

The governments in Kiev and Moscow have been locked in diplomatic clashes since Russia seized Crimea, and armed Russia-backed separatists have waged a deadly struggle in Ukraine’s east. More than 13,000 people have been killed during the conflict, a quarter of them civilians, according to the United Nations.

Mr. Zelensky called the prisoner exchange “the first step to stop the war.”

The plane carrying the sailors and 11 other prisoners took off around noon local time from Vnukovo Airport near Moscow, according to Ukraine’s presidency and Nikolai Polozov, a lawyer working on their case in Russia, who announced the news on the Ukrainian television channel 112. A similar plane departed from Borispol Airport near Kiev around the same time.

Oleg Sentsov, a Ukrainian filmmaker who had been sentenced to 20 years in prison in Russia on terrorism charges, was also included in the prisoner exchange, his lawyer Mark Feigin said on Twitter.

The prisoners returned to Russia included the journalist Kirill Vyshinsky, who was accused of treason in Ukraine; and Ukrainian soldiers who had deserted for Russia. Volodymyr Tsemakh, who is suspected of having taken part in the downing of a Malaysia Airlines flight over Eastern Ukraine in 2015 that killed 298 people, was also returned by Ukraine.

The Dutch government, which led the investigation into the crash, said that it deeply regretted his inclusion “under pressure from the Russian Federation.” Dutch investigators feared the release could jeopardize the investigation into Flight MH17. Mr. Tsemakh was released on bail by a Ukrainian court on Thursday.

The Russian Embassy in Kiev posted a picture on Twitter of the released Russians, writing: “Our people have been released! They are flying home.”

President Trump said on Twitter that the exchange was “very good news, perhaps a first giant step to peace.” Donald Tusk, the European Council president, said of the release, “Finally!” He wrote on Twitter, “I continue my call on Russia to release all political prisoners and respect Ukraine’s territorial integrity.”

Chancellor Angela Merkel of Germany on Saturday called the prisoner swap “a hopeful sign.” In a statement, Ms. Merkel called for progress on meeting the terms of the Minsk agreements, a 2015 cease-fire deal between Russia and Ukraine. President Emmanuel Macron of France has been pushing for a summit for leaders of Russia, Ukraine, Germany and France to discuss the crisis.

Ukraine’s president at the time of the sailors’ detention, Petro O. Poroshenko, had put his nation on a war footing with Russia as tensions over the shared Kerch waterway escalated into a crisis that dragged in NATO and the United Nations.

The sailors, who were onboard three small vessels, had been involved in a clash with a Russian tanker in the strait, a disputed waterway linked to the Black Sea. Some of them were wounded in a firefight.

In retaliation, Ukraine in July this year seized the Russian tanker that it said had been used during the confrontation. The tanker was accused of blockading the route under the Kerch Bridge, a structure built hastily and opened by Mr. Putin in 2018 as the first direct link from Russia to the Crimean peninsula.

Volodymyr Fesenko, the head of the Penta Center research institute in Kiev, said on Saturday that any optimism about a resolution to the overall crisis between the two countries should be moderate.

“I don’t exclude that it is just a tactical step for Putin to lure Zelensky into a more cooperative mode,” he said. “The ultimatums that Russia put forward, such as that Kiev should negotiate with the separatists directly and should give autonomy to the eastern regions, are just unacceptable to the Ukrainian government.”

Mr. Zelensky greeted the returned prisoners at the Kiev airport after their release. Standing near the plane, he said at a news conference: “On Aug. 7, we had a telephone conversation with Russian President Vladimir Putin. We have agreed on the first stage to unblock our dialogue and on the first step to stop the war: the release of our servicemen, our hostages, our sailors, our territories.”

“Next,” he added, “we will work on returning all our hostages and will continue working within the Minsk process on the disengagement of forces.”

Russia’s Foreign Ministry spoke favorably of the new Ukrainian leader. “In contrast to its predecessors, the new administration of Vladimir Zelensky showed its willingness to compromise,” it said in a statement.

Widespread reports of a prisoner exchange had been rumored for some time. This past week, at a forum in Vladivostok, in Russia’s Far East, Mr. Putin called the possibility of a prisoner exchange “a good step forward toward the normalization” of relations between the two countries.

Media outlets said Mr. Sentsov was moved from his Arctic prison last month to one in Moscow. And last week, when Ukraine’s chief prosecutor said on Facebook that the swap was underway, relatives of the detained sailors flocked to the Kiev airport. Their hopes were dashed when Ukraine’s presidency said that the exchange was not taking place.

Mr. Sentsov, detained in Crimea in 2014, had been accused of plotting to blow up bridges, power lines and a statue of Lenin. He denied plotting those actions but admitted opposing Russia’s takeover of Crimea. He was given a 20-year prison sentence in a penal colony in Siberia.

His imprisonment attracted international support and created an unwanted backdrop for Moscow as the men’s soccer World Cup got underway in 2018. That summer, he went on a hunger strike and was kept alive for 145 days by medical intervention.

The European Parliament’s president, David Sassoli, wrote on Twitter that Mr. Sentsov’s release brought “relief and profound joy.” The European Parliament last year awarded its prestigious human rights prize to Mr. Sentsov while he was in detention, elevating his prominence and embarrassing the Russian government.