"The jihadists have taken control of the headquarters building," used by the Kurdish military and civilian authorities, the Syrian Observatory for Human Rights said on Friday.
Fierce fighting had raged for the complex throughout the morning after IS militants captured part of it used by the Kurds' asayesh internal security force on Thursday.
"IS now controls 40 per cent of the town," after entering eastern districts on Monday and attacking from the west and the south, said the Britain-based Observatory, which has a wide network of sources inside Syria.
"The capture of the headquarters will allow the jihadists to advance on the border post with Turkey to the north of the town," Observatory director Rami Abdel Rahman said
"If they achieve that, they will have the Kurdish forces inside Kobani completely surrounded," he added.
UN warns of potential massacre
UN envoy Staffan de Mistura on Friday called on Turkey to allow Kurds to cross back into Syria to defend the key border town of Kobani from an onslaught by IS jihadists.
"We would like to appeal to the Turkish authorities in order to allow the flow of volunteers at least, and their equipment to be able to enter the city to contribute to a self-defence operation," de Mistura, the UN's Syria envoy, told reporters in Geneva.
He also called on Turkey, "if they can, to support the deterrent actions of the coalition through whatever means from their own territory."
Kobani, where Kurdish fighters are holding out after a three-week offensive by the IS militants, has become a crucial battleground in the fight against he IS extremists.
The statement marked an unusual one by the United Nations, which usually strives to stay neutral in conflicts, but de Mistura explained the rare appeal by the precarious situation in the key border town.
Kobani was "literally surrounded" except for one narrow entry and exit point, with up to 700 mainly elderly civilians still inside the city centre, with another 10,000-13,000 gathered nearby, he said.
"If this falls, the 700 plus perhaps if they move a little bit further the 12,000 people ... will be most likely massacred," he warned.
US-led aircraft pounded IS militants near Kobani on Thursday, but as fighting killed dozens, calls grew for ground action to support Kobani's beleaguered Kurdish defenders.
Since the IS assault on Kobani began in mid-September, nearly 500 people have been killed in and around the town and 300,000 have fled the region.
Battles have been raging especially fiercely since the jihadists breached Kobani's defences earlier this week.
Kobani, also known as Ain al-Arab, would be a major prize for the IS group, giving it unbroken control of a long stretch of the border.
The militants who have rampaged across large parts of Iraq and Syria have sent shockwaves through the international community as they have committed highly publicised brutal murders, including beheadings of several Western hostages.
