* Al Shabaab militants say behind attack

* Police say mortars hit airport, no one injured

(Adds al Shabaab)

By Ibrahim Mohamed and Mohamed Ahmed

MOGADISHU, April 13 (Reuters) - Militant Islamist insurgents fired mortars toward U.S. congressman Donald Payne as he left Somalia on Monday after a rare visit by a U.S. politician to the anarchic Horn of Africa nation.

The attack came a day after U.S. forces killed three pirates in an assault off Somalia that freed an American hostage.

The al Shabaab insurgents, whom Washington accuses of being al Qaeda's proxy in Somalia, claimed responsibility for the mortar attack, saying it was a message to the United States.

"We fired on the airport to target the so-called democratic congressman sent by (U.S. President Barack) Obama," Sheikh Hussein Ali, a spokesman for al Shabaab, told Reuters.

"Let him go back with the message of our strength and enmity towards the U.S and its allies. No single group can claim control of Mogadishu, and al Shabaab will continue its attacks."

Payne was unharmed.

Somalia's capital Mogadishu is one of the world's most dangerous places. U.S. officials and politicians have shunned travel to the battle-scarred city due to constant violence.

Somali Internal Security Minister Omar Hashi Aden told Reuters that Payne's plane was just airborne when the mortars fell. A police officer, however, said one mortar struck the airport before Payne left and five others impacted after.

Residents said insurgents and African Union (AU) peacekeepers exchanged fire after the attack, killing three civilians and wounding 12 other people near a market.

Ahmed, a local mechanic, said mortars came into the area where the insurgents had fired from. "I cannot make out if those injured ones are the civilians or the ones who fired," he said.

Payne spoke with the interim government's president and prime minister during the short visit.

He brought six bodyguards with him, and AU soldiers provided extra security.

VIOLENCE

Payne, 74, a New Jersey Democrat, defended the U.S. assault to free American ship captain Richard Phillips. Some fear it may stir more violence and anti-American sentiment.

"If there were no pirates, the U.S. government would not have intervened ... Every country has a right to defend its citizens," Payne told reporters before leaving.

Payne is chairman of the House of Representatives' Foreign Affairs Subcommittee on Africa and Global Health.

Jendayi Frazer, then the top U.S. diplomat for Africa, became the first high-ranking U.S. official to visit Somalia in more than a decade when she landed in Baidoa in April 2007.

She avoided Mogadishu because of violence there, preferring to meet officials in the provincial town that was then the seat of the Somali parliament.

Payne criticised Ethiopia's invasion of Somalia in late 2006, when Addis Ababa sent thousands of troops to crush an Islamist movement that had taken control of much of the south.

That attack ousted Sheikh Sharif Ahmed, then an Islamist leader in Mogadishu and now president of the government.

U.S. foreign policy toward the Horn of Africa nation has been haunted by the disastrous "Black Hawk Down" battle in Mogadishu in 1993 that killed 18 U.S. soldiers. (Additional reporting by Washington Bureau; Writing by Jack Kimball; Editing by Andrew Cawthorne and Jon Boyle)

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