Security sources told BBC Ukrainian eight more aircraft were badly damaged, while 20 service personnel could have been killed or injured.
The Morozovsk base houses Su-27 and Su-34 aircraft used on the front line in Ukraine, the sources said.
There has been no word from Russia on reports of an airfield attack.
The BBC has been unable to independently verify the reports.
However, Russian officials said that more than 40 drones had been aimed at the region, which borders Ukraine.
BBC Russian military analyst Pavel Aksyonov says an attack by so many drones over a short space of time could have overwhelmed air defences.
Meanwhile, four people were killed and over 20 wounded in Russian missile strikes on the Ukrainian city of Zaporizhzhia, Ukrainian officials said.
The eastern city of Kharkiv also came under air attack, with local authorities reporting two deaths from the Russian bombardment.
Russian strikes on Zaporizhzhia injured over 20, according to authorities
In Zaporizhzhia, regional governor Ivan Fedorov said several high-rise blocks and private homes were damaged.
Two missiles struck first, and then - after rescuers and police arrived - three more hit, he added. Ukrainian media said several journalists were among the injured.
Earlier, Ukraine's air force said that overnight and into Friday, Russia had attacked Odesa and Dnipropetrovsk regions and Kharkiv with 13 drones and five missiles.
All the drones were shot down, it said, but energy facilities were damaged in Odesa region, as well as residential buildings in Kharkiv.
Also, Russian troops continued their slow advance in Donetsk region. Ukrainian officials said the town of Chasiv Yar was coming under constant bombardment amid reports that Russian forces had reached the outskirts.
Reporting the Rostov attacks, the Russian defence ministry said Saratov, Kursk, Belgorod and Krasnodar regions had also been targeted. It said all the drones involved were shot down.
In Rostov's Morozovsk district, an electrical sub-station was hit, leaving about 600 people without power for a few hours, Rostov governor Vasily Golubev said on his Telegram channel.
He added that there had been some damage to a 16-storey residential block.
In a later post, he said eight people had been injured by explosives detonating on one of the fallen drones during an investigation of the site "not far from the aerodrome".
Ukraine has stepped up drone attacks on targets within Russia recently, targeting military and energy facilities.
It has warned repeatedly that its army is facing severe ammunition shortages, but has set a target of producing a million drones domestically this year.
On Tuesday it claimed responsibility for a drone attack in Tatarstan, more than 1,300km (807 miles) from the Ukraine-Russia border.
The strikes, targeting a drone factory and an oil refinery, are the deepest into Russian territory since the start of the war. Twelve people were reported injured.
bbc