The drone was downed near the village of Pratkunai in the Kaisiadorys district, about 100km west of the country's capital, Vilnius, and near the Kaunas Reservoir, Lithuania's National Crisis Management Centre said on Tuesday.
Lithuania's military said the incident marked the first time that a drone had been shot down in the country's airspace.
Authorities did not comment on the drone's origin or purpose, saying the incident needed further investigation.
The incursion follows a speech by Rutte on Monday at NATO headquarters in Brussels, where he said that strikes "close to NATO territory" showed Russian President Vladimir Putin's "desperation and also his desire to sow fear and terror".
"Russia is growing increasingly reckless as it continues its terrible war against Ukraine," Rutte said.
He also pledged to bolster the 32-nation alliance's own defences along its long eastern border with Russia.
The menace of drones has repeatedly loomed over the skies of Lithuania, which shares borders with Belarus, a key ally of Moscow, and the Russian exclave of Kaliningrad.
Antanas Matutis, commander of the Lithuanian air force, said that the wreckage had been found, and the drone "likely" was carrying explosives.
Vilmantas Vitkauskas, head of the crisis centre, told Lithuanian public broadcaster LRT that the drone most likely entered from Belarus.
It remained in Lithuanian airspace for about 30 minutes and flew along the outskirts of populated areas before being intercepted, he said.
In a separate encounter between Russian and NATO forces on Tuesday, Denmark's foreign ministry said a Danish air force helicopter had been fired upon with flares in connection with "routine photographic coverage" of a Russian frigate in waters off the port of Gedser in the Baltic Sea.
"This is completely unacceptable," the ministry wrote on X, saying that the Russian ambassador was summoned over the incident.
Meanwhile, Russian drone strikes continued across Ukraine following a social media post from US President Donald Trump on Monday that Russia and Ukraine had agreed "not to hit" each other's energy infrastructure.
Trump did not offer further details on the purported agreement, and Russia did not immediately comment on Trump's post.
Previous efforts to broker even a partial ceasefire have repeatedly fallen apart within hours, with both sides accusing each other of violations.
Ukrainian President Volodymyr Zelenskiy said late on Monday that Kyiv was "ready to support de-escalation" if the United States could ensure "Russia is genuinely ready to stop this war".
But Zelenskiy said on Tuesday that Moscow had continued to attack Ukraine's energy sector, logistics and critical infrastructure, and Kyiv had responded by striking an oil refinery in Syzran, a city in western Russia.
"There is no alternative to ending this war. And all forms of pressure on Russia must create the right diplomatic conditions," Zelenskiy said.
Ukraine's State Emergency Service said on Tuesday that Russian strikes on Kyiv had killed one person and wounded seven other people since the start of the day, with attacks carried out on the capital's gas stations.
Another two people were killed in separate Russian strikes in the Ukrainian cities of Sloviansk and Kramatorsk, Ukraine's State Emergency Service said.
Elsewhere, Russian forces shot down 222 Ukrainian drones overnight, Russia's defence ministry said.
A "massive" Ukrainian drone attack sparked several fires in the Russian city of Taganrog, regional governor Yuri Slyusar said.
He said on Telegram that two apartment buildings and three private homes had been damaged, as well as an educational institution, agricultural warehouses and a site belonging to Russian e-retailer Ozon.
Zelenskiy said on Tuesday that Kyiv's forces had targeted a drone production facility in the city.
