Baharin Kandal said IS fighters had retreated from all areas, except for two pockets of resistance in the east.
US-led air strikes have helped push back the militants, with another 14 conducted over the past 24 hours.
Meanwhile, the new UN human rights commissioner has called IS a "potentially genocidal" movement.
Zeid Ra'ad al Hussein described the group as the antithesis of human rights.'Tenuous'
Speaking by phone, Kurdish commander Baharin Kandal told the BBC's Kasra Naji that she hoped the city would be "liberated soon".
Ms Kandal said her militia group had been receiving arms, supplies and fighters but she refused to say how, reports our correspondent, who is on the Turkish border near Kobane.
At the scene: BBC's Kasra Naji on the Turkey-Syria border
Kurdish defenders have victory in their sights. After exactly a month of fighting, they say they have driven Islamic State from most of the city.
But from a hilltop across the border in Turkey, it is clear there is still fighting going on, particularly in the north of the city. Small and heavy arms fire can be heard, as well as occasional explosions. There have also been several air strikes this afternoon by the US-led coalition.
One 32-year-old Kurdish militia commander, who leads the fighting in the east of the city, told me she hoped the city would be "fully liberated" very soon.
Her comments reflect an air of optimism here in Turkey among the Kobane refugees who are hoping to go back to their town in the next few days.Major test
The battle for Kobane, which is also known as Ayn al-Arab, is regarded as a major test of whether the US-led coalition's air campaign can push back IS.
US Central Command said that bomber and fighter aircraft had conducted 14 air strikes on Wednesday and Thursday, all of them targeting IS around Kobane.
The strikes "successfully struck 19 IS buildings, two IS command posts, three IS fighting positions, three IS sniper positions, one IS staging location and one IS heavy machine gun", a statement said.
It said the air strikes had "continued to slow IS advances, but that the security situation on the ground in Kobane remains tenuous".
A Kurdish official in Kobane, Idris Nassen, confirmed to the AFP news agency that IS had pulled back from some areas and that "the international coalition has fought IS more effectively during the last few days".
But he warned: "We need more air strikes, as well as weaponry and ammunition to fight them on the ground."New propaganda film
Meanwhile, IS militants released another propaganda video featuring British hostage John Cantlie on Thursday.
The journalist, who was kidnapped in Syria in 2012, has appeared in several videos released by IS that have all followed the same format - with Mr Cantlie addressing the camera from behind a desk.
There are no signs of violence in the videos but Mr Cantlie has made it clear he is speaking as a prisoner whose life is in danger.
Earlier, Zeid Ra'ad al Hussein, the new UN human rights commissioner, described IS as the antithesis of human rights and "a diabolical, potentially genocidal movement".
He said: "The way it has spread its tentacles into other countries, employing social media and the internet to brainwash and recruit people from across the globe, reveals it to be the product of a perverse and lethal marriage of a new form of nihilism with the digital age."
Activists say more than 600 people have been killed since the jihadist group launched its assault on Kobane a month ago.
More than 160,000 people have fled the mainly Kurdish town in the face of the IS advance.
Capturing the town would give the group unbroken control of a long stretch of the Syrian-Turkish border.
IS fighters, who have seized large areas in Syria and Iraq, have gained a reputation for brutal tactics, including mass killings and beheadings of soldiers and journalists.