Zabihullah Mujahed, a Taliban spokesman, said that the movment had carried out the suicide attack, about 30km from the border with Pakistan.
Mohammad Bilal, a secretary to the Khost governor, said: "It was a suicide car bombing against the gates of the Salerno camp.
"Moments later a second car bomber came and wanted to detonate his bombs. Police identified him and opened fire on him. He came out of the car and escaped on a motorcycle."
"They wanted to disturb Independence Day," he said.
Independence day warning
Shortly before the attack, the US and Nato's International Security Assistance Force (Isaf) had warned of possible attacks as Afghanistan marked its independence from Britain 89 years ago.
Major General Jeffrey J. Schloesser, the senior US commander in Afghanistan, said that "credible intelligence" indicated that Taliban fighters planned to launch attacks at the independence celebrations.
"We recommend to all Afghans to be vigilant at large public events and other locations where crowds gather and report suspicious behaviour to security forces," the US military said in a statement.
Kabul was locked down with 7,000 police on patrol and checkpoints at nearly every city centre intersection as well as main entry points into the capital. Even the location of the official celebration was kept secret for security reasons.
Reacting to the suicide bombing in Khost, Hamid Karzai, Afghanistan's president, said that by killing "innocent civilians on Independence Day, the terrorists showed their hostility to the freedom of Afghan people".
Later on Monday he laid a wreath at the Minar-i-Istiqlal (Column of Independence) in the defence ministry grounds in Kabul during a small and tightly secured ceremony.
The Taliban, who were driven from power in a US-led invasion in 2001, released an Independence Day statement saying that Afghanistan was again under the "occupation" of "cruel crusaders".
Meanwhile, an Afghan woman and two children were killed when British soldiers fired rockets at a compound in southern Afghanistan over the weekend, the British military said.
Another four civilians were hurt in the incident on Saturday in the southern province of Helmand, a British military statement said.
"An Afghan woman and two children died in the Sangin area when UK forces fired rockets at Taliban fighters, unaware that civilians were in the vicinity of enemy fighters in a local compound," it said.
One of the wounded, a child, was being treated in a French-run hospital in Kabul and the three others were in a hospital at the main British base in Helmand.