Most accounts of Maryinka, southwest of the Russian-held regional centre of Donetsk, describe it as a ghost town.
Putin said control of the town, which was once home to 10,000 people, will allow the Russian forces to move enemy combat units away from Donetsk.
"Our troops (now) have the opportunity to reach a wider operational area," he said in a video of the exchange between him and Shoigu posted online by a Kremlin journalist.
Russia's last major success on the battlefield was the capture in May of Bakhmut, theatre of some of the bloodiest fighting.
Ukraine launched a counteroffensive in June aimed at retaking land in the country's south and east, including Bakhmut.
Ukraine's forces have made little progress in the counteroffensive in the face of entrenched Russian resistance.
The Ukrainian army has rejected reports that Russian forces have seized the town of Maryinka. (EPA PHOTO)
Russian troops have also intensified land and air-based attacks on the nearby town of Avdiivka since mid-October as the focal point of their slow-moving push through eastern Ukraine's Donbas region in the 22-month-old conflict.
Avdiivka was briefly captured in 2014 by Russian-backed separatists who seized large chunks of eastern Ukraine.
Fortifications were later built around the town - seen as a gateway to Donetsk.
"Ukrainian defence forces continued to hold back the enemy in the areas of Maryinka and Novomykhailivka in the Donetsk region, repelling five Russian attacks," Ukrainian General Staff said in its dispatch early on Monday.
Later in the day, the Ukrainian army denied that Russian forces had seized Maryinka.
"It's not correct to talk about seizing Maryinka," Ukrainian military spokesman Oleksandr Shtupun told Ukrainian television.
"Our forces are within the city," he said.
Meanwhile, a draft law posted on Ukraine's parliament website late on Monday proposes lowering the age of those who can be mobilised for combat duty from 27 to 25.
President Volodymyr Zelenskiy told his end-of-year news conference on December 19 that the military had proposed mobilising 450,000-500,000 more Ukrainians, but that it was a "highly sensitive" issue that the military and government would discuss before deciding whether to send the proposal to parliament.
Zelenskiy said he wanted to hear more arguments for mobilising additional people. "This is a very serious number," he said.
Ukraine's troop numbers are not known, but in the past it has been said the country has around 1 million people under arms.
US officials estimate that hundreds of thousands have been killed and wounded since Russia invaded Ukraine. Neither country publishes its casualty figures.
