Mr. Talabani and the small delegation that Kurdistan has in Washington have used tactics similar to those of corporations that spend millions of dollars to grease the levers of power, retaining five firms to push its cause. They have been effective, winning over a rare combination of military hawks, conservative Republicans and a collection of liberal Democrats in Congress. More important, they secured a commitment late last month for $415 million in additional aid to support the Kurds’ pesh merga militia force.

“They are willing to fight,” said Representative Gerald E. Connolly, Democrat of Virginia, who is a member of the Kurdish-American Congressional Caucus, which the lobbying team helped set up in the House. “They are the only group that has had consistent battlefield success.”

Kurdish officials say they will continue to press for additional money for ammunition, armored vehicles and protective gear against chemical weapons, requests that lawmakers say they are looking for ways to accommodate.

What is perhaps more notable is that the Kurds are succeeding even in a climate of austerity on Capitol Hill. The relative stability in Kurdistan and in its capital, Erbil, contrasted with the continuing chaos in Baghdad — highlighted by the storming last Saturday of the Iraqi Parliament building by protesters — has only enhanced the clout of the Kurds in Washington. But it is also an area of a continued threat, including to American military forces there, with one American service member killed by enemy fire near Erbil on Tuesday during a clash with the Islamic State.