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Romanians living close to the border with Ukraine received emergency alerts early on Wednesday urging them to remain calm and take shelter as Russia hit Ukrainian ports across the Danube.
The defence ministry said it had found Russian drone parts on Romanian territory for the third time in a week.
But the Nato member's government has told the residents the risk of a cross-border escalation was minimal.
Makeshifts concrete shelters are being set up by the Romanian military.
Shortly after midnight local time, residents in eight villages in eastern Romania started receiving messages on their phones that warned of "the possibility of falling objects from the surrounding airspace".
"Keep calm! Take shelter in basements or in civil protection shelters," it said. "In the absence of a shelter, stay inside the house, away from the windows and exterior walls."
Romania's defence ministry said it had detected a group of drones heading towards Ukrainian ports, just across the Danube from Romania, that Russia has increasingly targeted in recent weeks.
It said the warning had been issued because of "possible cases of drone impact" in areas around Tulcea and Nufaru. The alert was lifted at 05:00 local time (03:00 BST), it added.
Residents in border villages say they can often hear drones flying over Romanian territory, although Nufaru is 14km (8.7 miles) south of the border.
"When the sirens wail at night, that's when you should expect the drones to come," Costica Tanase, a resident in Plauru on the Ukraine-Romania border told the AFP news agency.
"It shakes like an earthquake" when a drone approaches, 70-year-old farmer Gheorghe Puflea said.
Romania has condemned Russian attacks on grain infrastructure in the Ukrainian ports of Izmail and Reni as "unjustified and in serious contradiction with the rules of international humanitarian law".
Its defence ministry also announced it had found fragments that "are from a drone similar to those used by the Russian army" on a hill 2km (1.25 miles) south of Nufaru.
"It's hard to see the area from the ground. You can see it better from the air. That's why the crash site was first spotted from a helicopter," local military commander Antoanel Vatamanu said.
Romania initially rejected reports last week that Russian drone parts could have landed on its territory, but the authorities have now made three separate discoveries of possible fragments and are investigating their origins. The latest find is the furthest from the border with Ukraine.
Speaking before Wednesday's discovery, Romanian Prime Minister Marcel Ciolacu said the country was not being targeted: "Nobody is attacking us, folks! Let's calm the nerves of the population, or else we will all go crazy."
"We just have some wreckage from a drone that was hit by the Ukrainian army," he said.
The authorities have started building two makeshift concrete structures residents in the border area can shelter in.
Around 50 soldiers were using concrete slabs to build 10m (32.8ft) long, 2m (6.5ft) wide and 1.5m (5ft) high shelters, the defence ministry said.
Raed Arafat, the head of the emergency services in the area, said the measure was a precaution: "Even though the risk could be around 0.001%, we still need to be prepared and to protect the inhabitants."
"Inhabitants felt unsafe and and we tried to do something for them," said Mr Ciolacu.
Russia's bombing campaign against Ukraine's Danube ports - in close proximity to Nato members such as Romania - has raised fears the war in Ukraine could escalate beyond its western borders.
The Ukrainian military said it had shot down 32 drones in the latest attack.
"A total of seven civilians were injured as a result of Russian attack drones in the Izmail district, six people in Reni and one in Izmail," Odesa regional military administration head Oleh Kiper said.
The ports are a vital export route for Ukraine, in particular since Russia pulled out of a UN-brokered deal that allowed safe grain shipments via the Black Sea in July.