The ministry tweeted: "Ras al-Ayn's residential center has been taken under control through the successful operations in the east of Euphrates" River.
But Kurdish authorities denied the town had fallen to the Turks, saying fighting was continuing.
A White House statement announcing the withdrawal of USA troops prior to the Turkish offensive said that Turkey would now become responsible for all ISIS fighters in the region-a suggestion that was widely criticized.
The Syrian Observatory for Human Rights, a UK-based organization which reports on the war, said Turkey-backed groups had killed a total of nine civilians on the road, including Hervin Khalaf, secretary general of the Future Syria Party.
The incursion, which began after US President Donald Trump withdrew his troops who had been fighting alongside Kurdish forces against Islamic State jihadists, has opened a new front in the eight-year Syrian civil war and drawn fierce global criticism.
France's President Emmanuel Macron has discussed the Turkish offensive in Syria with Mr Trump, and warned about a possible resurgence of Islamic State activity as a result of the military action.
The Syrian Democratic Forces, or SDF, called on the United States to carry out its "moral responsibilities" and close northern Syrian airspace to Turkish warplanes.
Iran's call came on the fourth day of Turkey's offensive against the Syrian Kurdish YPG militia, which Turkey's Government believes has links to the PKK.
It also plans to resettle more than three million Syrian refugees in the zone who are now in Turkey.
Turkey's Defence Ministry subsequently said Ras al Ain had been brought under control, with officials posting photos showing deserted streets and Syrian rebels standing on Kurdish militia flags.
The premier during the conversation told the Turkish president that "Pakistan fully understands Turkey's concerns relating to terrorism", a statement issued by the Prime Minister's Office said.
Washington said Turkey's incursion was causing "great harm" in relations with its North Atlantic Treaty Organisation ally. Erdogan threatened to send them to Europe if the European Union did not back his assault.
Turkey considers Kurdish militants in northern Syria a "terrorist" offshoot of Kurdish insurgents in its own territory. "U.S. forces have not withdrawn from Kobani", it said.
The SDF lost 11,000 fighters in the protracted US-led campaign against the Islamic State group before finally overrunning its self-proclaimed "caliphate" in March.
Upwards of 100 women affiliated with ISIS also started a major disruption at the Al-Hol camp near the Iraqi border-setting tents ablaze and attacking security guards with sticks and stones-and video footage of the incident showed roughly 20 women running through an open field in an attempt to flee.
Later at the Pentagon, a USA official told several news outlets that the source of the shelling in Kobane had not yet been confirmed.
Turkey and its allied fighters began the offensive on Wednesday to push back the Kurdish Peoples' Protection Units (YPG), the backbone of the SDF, from its border. The Kurdish militia known as the People's Protect Units, or YPG, has formed the backbone of USA efforts in northeastern Syria and collaborated with US troops, but Turkey considers the group to be part of a Kurdish movement, the PKK, that it deems a terrorist organization.