Outgunned Kurdish militia were struggling to prevent the jihadists closing off the last escape route for the thousands of civilians still in or near the town, prompting an appeal for urgent military assistance.
US-led warplanes have intensified air strikes against IS fighters who have been attacking Kobane for three weeks but the Pentagon has warned that, without a force on the ground to work with, there are limits to what can be done.
The jihadists' advance has brought the front line to just 1.3 kilometres from the border.
"IS now controls 40 per cent of the town," said the Britain-based Syrian Observatory for Human Rights.
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"The capture of the headquarters will allow the jihadists to advance on the border post with Turkey to the north of the town," its director Rami Abdel Rahman told AFP.
"If they achieve that, they will have the Kurdish forces inside Kobane completely surrounded."
Kobane was "literally surrounded" except for one narrow entry and exit point to the Turkish border, de Mistura said.
Up to 700 mainly elderly civilians were still inside the city centre, and another 10,000-13,000 gathered nearby.
"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.
"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," he told reporters in Geneva.