The British-based Syrian Observatory for Human Rights said the militants carried out dawn raids on rural villages inhabited by the ancient Christian minority west of Hasaka, a city mainly held by the Kurds.
Syrian Kurdish militia launched two offensives against the militants in northeast Syria on Sunday, helped by U.S.-led air strikes and Iraqi peshmerga.
This part of Syria is strategically important in the fight against Islamic State because it borders territory controlled by the group in Iraq, where it last year committed atrocities against the minority religious Yazidi community.
Many Assyrian Christians have emigrated in the nearly four-year-long conflict in which more than 200,000 have people have been killed. Before the arrival of Kurds and Arab nomadic tribes at the end of the 19th century, Christians formed the majority in Syria's Jazeera area, which includes Hasaka.
Sunday's offensive by Kurdish YPG militia reached within 5 km (3 miles) of Tel Hamis, an Islamic State controlled town south east of Qamishli, the Observatory said.
At least 14 IS fighters died in the offensive, which involved Assyrian fighters, and eight civilians were also killed in heavy shelling by the Kurdish side which seized several Arab villages from Islamic State control, the monitoring group said.
Last year, Islamic State fighters abducted several Assyrians in retaliation for some of them fighting alongside the YPG. Most were released after long negotiations.
Military experts following Syria said Islamic State was trying to open a new front to relieve pressure on the group after the string of losses since being driven from the Syrian town of Kobani near the border with Turkey.
"Islamic State are losing in several areas so they want to wage an attack on a new area," said retired Jordanian general Fayez Dwiri.
Since driving IS from Kobani, Kurdish forces, backed by other Syrian armed groups, have pursued the group's fighters as far as their provincial stronghold of Raqqa.
A resident of Hasaka, which is jointly held by the Syrian government and the Kurds, said hundreds of families had arrived in the last few days from surrounding Christian villages and of Arab Bedouin refugees were arriving from areas along the border.
"Families are coming to Hasaka seeking safety," said Abdul Rahman al-Numai, a textile trader said via telephone.
(Reporting by Suleiman Al-Khalidi; Editing by Raissa Kasolowsky)