The statement from Iran's army headquarters comes just hours after Americans elected their next president, Barack Obama, who has said he would toughen sanctions on Iran but has also held out the possibility of direct talks with Tehran.
The warning also followed a raid by U.S. forces into Syria last month, a move that was condemned by Damascus and Tehran.
"In the statement, it says recently it has been seen that American army helicopters were flying a small distance from Iraq's border with Iran and, because of the closeness to the border, the danger of them violating Iran's border is possible," state radio said, quoting the army statement.
"Iran's armed forces will respond to any violation," it said.
Washington, which has not had diplomatic ties with Tehran since 1980, has accused Iran of funding, equipping and training militants in Iraq. Iran denies this and says instability is the due to the presence of U.S. troops who should quit Iraq.
The United States also accuses Iran of seeking to build nuclear weapons, prompting its push to isolate the Islamic Republic with more U.S. and U.N. sanctions.
Iran, the world's fourth largest oil producer, says it wants atomic technology to make electricity and save its oil and gas for export.
Syria says eight civilians were killed in the attack on October 26, which occurred close to Syria's border with Iraq and which Damascus has condemned as "terrorist aggression" by the United States.
Washington has yet to confirm officially the raid but officials have said on condition of anonymity that the operation is believed to have killed a major al Qaeda militant responsible for smuggling foreign fighters into Iraq.
(Reporting by Zahra Hosseinian and Parisa Hafezi, writing by Edmund Blair)
