One attacker drove a car rigged with explosives and blew himself up outside the gates.
Message to Petraeus
Omar said a second attacker struck inside the base.
"The first suicide attacker detonated at the entrance, and the second detonated inside the premises, killing one foreign national," he said.
Foreign workers inside the compound fled to the roof to escape the fighting.
A spokesman for the Taliban told Al Jazeera that six men took part in the attack, describing it as a "welcome" for General David Petraeus, the new commander of US and Nato forces in Afghanistan.
Petraeus arrived in Kabul on Friday.
Al Jazeera's Zeina Khodr, reporting from the Afghan capital, said the Taliban's message to Petraeus was that he should start planning the withdrawal of foreign forces from the country.
"But Petraeus is warning of tough fighting in the months to come," she said.
"Really his main test will be in Kandahar if that much-awaited [security] operation begins.
"It's the stronghold of the Taliban. Clearing that area and providing better services for the people there - that will be his test."
|The Taliban said the Kunduz attack was a "welcome" message for Petraeus [AFP]
Reacting to Friday's attack in Kunduz, Captain Jane Campbell, a spokesman for Nato, said."This attack shows the insurgents' desire to prevent progress, and draws attention to their true goal of serving themselves rather than the people of Afghanistan."
DAI runs two aid programmes in Kunduz, according to the US Agency for International Development.
One programme provides grants for small businesses while the other works with farmers to improve agricultural technology.
The Taliban accused the company of providing intelligence and support to US troops.
Attacks on foreign aid workers are common in Afghanistan.
A 2009 report from the London-based Overseas Development Institute found that Afghanistan was one of the three most dangerous countries in the world for aid workers.