The town has borne the brunt of Russian advances in recent months and is a stepping-stone to the key logistical hub of Pokrovsk.
Ukraine has not acknowledged the fall of Kurakhove, which is 35km (21 miles) south of Pokrovsk.
Fierce fighting has also been under way in Russia's Kursk region in recent days after Ukraine launched a counter-attack on Sunday.
An image taken from social media and supplied by the Reuters news agency appears to show a soldier holding up a Russian flag in Kurakhove. The image has not been verified by the BBC.
BBC