The attack is alleged to have taken place from inside Turkey.
The struggle started when an armored vehicle was detonated by a suicide bomber as the vehicle approached a crossing near the border with Turkey.
The radical militant group, which is also referred to as ISIS attacked Kobani on all sides, a spokesperson for the Democratic Union Party of the Kurds, Nawaf Khalil said to news wire agencies.
Approximately 40 people on each side had been killed in the fighting on Saturday.
The Islamic State has fought to gain control of Kobani for the past two months as the organization looks to build its caliphate. Already, ISIS has take control of wide swaths of land inside Iraq and Syria.
A number of questions were raised about the siege, most notably whether the efforts by the U.S. to aid the fight against the ISIS is working, but more importantly about whether the country of Turkey was involved with Islamic State, which Turkey has denied.
Turkey has been hesitant in helping Kobani because of not wanting to incite a Kurdish national movement that would cross over into its country.
A number of reports have said that the attack had been launched from inside Turkey, but Ankara said the reports had been inaccurate.
The government of Turkey released a statement that said the claims the vehicle had reached the border gate via crossing into Turkish soil had been a lie.
However, there were witnesses inside Kobani that reported seeing attacks from the side of the Turkish border.
One activist said he saw fighters from Islamic State inside grain silos on the border side of Turkey launching attacks toward the border crossing.
The U.S. has a led a coalition that are using airstrikes in an attempt to slow down the movement of ISIS.