"Since Thursday last week, we believe that 42,000 Syrians have now swept across Syria's far northeast border into Iraq," said Adrian Edwards, spokesman for the UN high commissioner for refugees.
On August 15, Iraq's autonomous Kurdistan region opened a crossing on its Tigris River border with Syria, which previously had been controlled tightly amid fears of a spillover of conflict.
Consequently, there was a massive influx of refugees, most of them Syrian Kurds.
The biggest influx was last weekend, and aid agencies said that Iraqi Kurdistan had initially imposed a daily limit of 3,000, but it was not if this remained in place.
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"Clearly, the principle has to be that people who need a safe haven have access to it," Edwards told reporters.
He said it was tough to say how many more refugees might be on their way.
"The numbers have been coming down, and we expect that trend to continue. But it's obviously contingent on the situation across the border in northern Syria itself," he explained.
Around two million Syrians have fled their homeland since the revolt against President Bashar al-Assad erupted in 2011.
Most have found a haven in Lebanon, Jordan and Turkey, while UN figures put the number in Iraq at least 1,97,000, including the latest influx.
Syria's long-marginalised Kurdish minority had tried to avoid antagonising forces loyal to Assad or rebels fighting to overthrow him.
But there has been fierce fighting in recent weeks between Syrian Kurdish forces and the Islamist Al-Nusra Front, which is also fighting Assad.