BAGHDAD — Kurdish fighters opened offensives against Islamic State militants in several parts of northern Iraq on Tuesday, seizing control of a border crossing with Syria that has been a major conduit for the insurgents, officials said.

In a predawn push, pesh merga forces of the Kurdistan Regional Government fought their way into the Rabia district, near the Syrian border, seizing control of two villages by late morning and, by day’s end, the border crossing, Kurdish officials said.

The Islamic State, also known as ISIS, had controlled Rabia since early June, when jihadist fighters swept across the border from Syria and quickly overwhelmed Iraqi security forces throughout the region, including in Mosul, Iraq’s second-largest city.

The militants, who have declared an Islamic caliphate stretching across eastern Syria and western Iraq, have used a highway between Rabia and Mosul, 70 miles away, to transport fighters, weapons, armored vehicles and supplies freely between the two countries.