The news prompted celebrations among residents who fled across the frontier into Turkey, with thousands gathering at the border and hoping to return, more than four months after the fighting began.
The town's recapture marked a key symbolic and strategic blow against IS, but officials warned massive reconstruction was needed and the fight would continue for the surrounding villages.
The Kurdish People's Protection Units (YPG) announced the "liberation" of Kobane on Monday, depriving IS of a strategic prize to add to its territory in Syria and Iraq.
"Our forces fulfilled the promise of victory," the militia said, cautioning that fighting was not over yet.
"The process to ultimately liberate Kobane canton (region) is ahead of us. We pledge that we will successfully carry out this promise as well."
There was fighting in villages around the town on Tuesday, both to the southeast and the southwest, according to the Syrian Observatory for Human Rights, a Britain-based monitor.
Kobane activist Mustafa Ebdi said the US-led coalition fighting IS carried out fresh air strikes around the town on Monday evening and Tuesday morning.
In Turkey, thousands of Kurds among the 200,000 who fled Kobane and the surrounding area, flocked to the border.
Most went to celebrate, but some tried to cross the frontier, which remains officially closed.
Turkish security forces used tear gas and water cannon to push back those who approached the barbed wire separating the two countries.
Only a handful of people were able to cross, including Idris Nassan, deputy foreign minister for the Kobane regional government.
"People are very glad. They are celebrating. Morale is very high," he told AFP from the town.
He said the regional government was urging residents not to return yet.
"There is massive destruction. At least 50 percent of the city is destroyed," he said.
Signup to our newsletter and follow us on Facebook and Twitter!
"We are asking them to wait and not come immediately because we don't have basic necessities for them. There is no food, no medicine. We don't have electricity or water."
Nassan said the regional government would now appeal to the international community for help.
"We need aid. We need experts for reconstruction. We also need weaponry to continue to fight," he said.
"This is the first stage, the liberation of Kobane. The next stage is the liberation of the villages."
'BIG BLOW' AGAINST IS
As the Kurdish militia raised their flags over Kobane, Turkish President Recep Tayyip Erdogan said his country opposed the idea of a Kurdish-controlled autonomous government in northern Syria.
"We do not want a new Iraq. What's this? Northern Iraq," Erdogan told Turkey's Hurriyet newspaper.
"A northern Syria there after northern Iraq... It is not possible for us to accept this," he said.
The loss of Kobane appeared to be a major blow for IS, which had seemed poised to seize the town after it began its advance on September 16.
It lost nearly 1,200 fighters in the battle, of a total of 1,800 killed, despite outgunning YPG forces with sophisticated weaponry captured from Iraqi and Syrian military bases.
Analysts said air strikes by the US-led coalition had been key to the YPG's success, taking out some of the jihadists' heavier weaponry and hitting their supply routes.
Forces from the Iraqi Kurdish peshmerga, as well as some Syrian Arab opposition forces, also reinforced the YPG in the fight.
IS emerged in Syria in 2013, and quickly captured large areas there and in neighbouring Iraq, imposing its harsh interpretation of Islamic law.
But analysts said the loss of Kobane could put the brakes on its plans for expansion.
"Despite all that manpower, all that sophisticated weaponry, IS couldn't get the city, so it's a big blow for their plans and it's a great achievement for the Kurds," said Kurdish affairs analyst Mutlu Civiroglu.