The drone was downed overnight 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.
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 didn't 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".
"We see that the Russians feel the pressure from our precise strikes on Russia's war economy. And so far, no one has seen any genuine desire on Russia's part to move toward ending the war," he said.
"But if our American partners can ensure that Russia is genuinely ready to stop this war aimed at destroying life - and to do so for real and for the long term - Ukraine will take its own de-escalatory steps."
Ukraine's State Emergency Service said Tuesday that eight people were wounded in Russian strikes on Kyiv since the start of the day, including two who were wounded in strikes on the capital's gas stations.
Elsewhere, Russian forces shot down 222 Ukrainian drones overnight, Russia's defence ministry said on Tuesday.
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.
