KABUL, Afghanistan — The Taliban claimed to have captured the capital of the western province of Farah on Tuesday, while government officials and their American military backers vowed that the authorities would quickly oust insurgents from the city, the first to be overrun by the extremist militants in two years.

Only the provincial governor’s compound remained in government hands after a long day of fighting, which continued into the night, according to numerous residents and some local officials, as well as the Taliban insurgents. Gov. Basir Salangi fled the city after the insurgent attack began around 2 a.m., but he remained in Farah Province at a military base a few miles outside Farah city, according to numerous local officials.

The fighting in Farah was part of a recent increase in the tempo of attacks by the insurgents, since their announcement of a spring offensive late in April and their explicit rejection of Afghan government peace initiatives.

A senior Afghan police official, reached by telephone inside Farah City, described the government’s situation as “out of control” and predicted the insurgents would renew their offensive under cover of dark, when air support was less effective against them. The official spoke on condition of anonymity because he was contradicting official government reports.

Qari Yousef Ahmadi, a spokesman for the Taliban, issued a statement late Tuesday afternoon on WhatsApp messaging groups claiming that the insurgents had taken “most parts of Farah city.” He said that three senior Afghan security officers were among dozens killed in the insurgents’ assault, which government officials denied.

Afghan officials confirmed that a district police chief in Farah died of his wounds Tuesday, but they denied claims that the deputy provincial police chief had also died, saying he was critically wounded. Other claims by the insurgents were exaggerated, officials said.

“They kill one, and they claim three,” said Gen. Mohammad Radmanish, spokesman for the Ministry of Defense

The American military said in a statement that it had A-10 warplanes over the city, “which remains under government control.” Those aircraft are generally used for close support of troops on the ground, but the statement did not clarify whether they were joining Afghan military helicopters in bombing the Taliban. “U.S. Forces-Afghanistan stand firm beside our Afghan partners and will continue to reinforce this Afghan-led offensive,” the statement said.

The offensive by all accounts was launched by the Taliban side about 2 a.m. on Tuesday, with attacks on the defensive perimeter of the city, capital of a province that has seen heavy fighting for several months. By midday, Taliban insurgents were posting photographs and video of themselves celebrating in the main traffic circle at the center of the city, and parading captured Afghan military vehicles and weapons.

Farah, a city of 50,000, would be the second Afghan city to fall to the insurgents since the war began. The northern city of Kunduz fell twice to the Taliban, in 2015 and 2016, but was retaken by the government both times.

In Kabul, Afghan officials said they would quickly expel the Taliban from Farah. “We are hoping to get control of the fighting in the city,” Gen. Radmanish said at a news conference. “Our reinforcements are trying to use their capabilities to overcome the situation.”

By the end of the day that had not happened, but Gen. Radmanish insisted the center of the city remained in government hands. “The governor’s office, the police headquarters, the National Directorate for Security, the university and the airport are under our control,” he said. “We are using all of our capabilities to change the situation there.”

Residents reached by telephone inside Farah late Tuesday all said the insurgents appeared to control most if not all of the city, other than the governor’s offices. “It is amazing how easily the Taliban managed to enter the city,” said one resident, Dill Agha. “They are telling residents to remain in their homes, that they are not hurting civilians or any military or police who surrender themselves.”

Fareed Bakhtawar, head of the Farah provincial council, said that he had joined volunteers to fight the Taliban but that the insurgents had taken large areas of the city. “There is no extra force to help us, and the situation is getting worse,” Mr. Bakhtawar said.

Fighting has been intense for months in the western province — much of which, outside of military bases, has already fallen under Taliban domination — with hundreds of police officers and soldiers killed. The fighting has increased in Farah because of Taliban reinforcements sent from neighboring Helmand Province, where the insurgents have fared less well in recent months, according to Afghan officials.

Qasim, a resident of the city who worked for a foreign organization, said he and friends tried to flee but were advised to hide in their homes because the insurgents were checking identification in order to find people like him. “The fighting is very fierce and it is very dangerous to go out,” he said. “All the people we know are trapped in their homes. We were told not to even peek outside.”

A Taliban spokesman, Zabihullah Mujahid, also posting on WhatsApp, told Farah residents not to worry. “Everyone should stay home, be calm, and pray,” he said. “The mujahedeen are your brothers, don’t pay attention to the propaganda of the enemy.”

As they advanced through the city on Tuesday, the Taliban posted numerous videos on their Facebook account and on unofficial Twitter accounts showing checkpoints and government buildings purportedly falling to their control in rapid succession throughout the day. They were also seen both on social media and by residents driving commandeered police and military vehicles.

A spokesman for President Ashraf Ghani, Shahhussain Murtazawi, also denied that the city had fallen. “Reinforcement forces have arrived in Farah City, the situation is changed there, it’s not as tense as it’s described in social media,” he said.

An Afghan Air Force general said that helicopter gunships from Shindand Air Base in western Afghanistan were joining the fight, targeting Taliban positions in the city. The general, who spoke on condition of anonymity because he was unauthorized to speak to the news media, said the air force was hampered by the need to avoid bombing residential dwellings.

The videos posted on the Taliban’s social media accounts showed Afghan Army vehicles and weaponry that they claimed to have seized in Farah. In one Facebook post, the insurgents showed an Afghan police officer, his hands bound behind his back, while an argument ensued among them over whether to kill him or not. It was unclear what happened to the officer.

In the past, when Kunduz fell to the Taliban, American air power and Special Operations troops intervened to help restore the city to government control. It was not clear if that was contemplated in Farah, but the American military’s statement indicated authorities were watching developments closely. “As we have seen over the last couple of days,” the statement said, “the Taliban are unable to hold terrain during such isolated attacks and their unsuccessful raid attempts on district centers in Badakhshan, Baghlan, Faryab and Zabul.”

Fighting in those and other places have cost the lives of more than 100 soldiers and policemen in the past week, and the insurgents have managed to hold onto the district of Tala Wa Barfak, for a week now, cutting a strategic highway in northern Afghanistan’s Baghlan Province since May 8.